Notes to a video lecture on http://www.unizor.com
Metric Invariance
One of the most important characteristic of space we live in is distance between two points.
Classical approach to Physics prior to Theory of Relativity was that our space is Euclidian and, therefore, the square of a distance between two points
A(XA,YA,ZA) and B(XB,YB,ZB)
equals to
d²(A,B) = (XB−XA)² +
+ (YB−YA)² + (ZB−ZA)²
Let's analyze how this distance transforms by Galilean transformation.
Assume, we have two reference frames α{T,X,Y,Z} and β{t,x,y,z}.
Further assume that time is absolute and the same in both reference frames (t=T), that frame β at time t=T=0 coincides with α and then moves relative to α along a straight line with uniform velocity vector v(vX,vY,vZ), preserving parallelism of axes.
Then the Galilean transformation of coordinates from α to β, as specified in the previous lecture, is
t = T
x(t) = X(T) − vX·T
y(t) = Y(T) − vY·T
z(t) = Z(T) − vZ·T
Applied this transformation to coordinates of points A and B, we obtain
xA(t) = XA(T) − vX·T
yA(t) = YA(T) − vY·T
zA(t) = ZA(T) − vZ·T
and
xB(t) = XB(T) − vX·T
yB(t) = YB(T) − vY·T
zB(t) = ZB(T) − vZ·T
Now we are ready to calculate the distance between these two points in the reference frame β{t,x,y,z}.
dβ²(A,B) = (xB−xA)² +
+ (yB−yA)² + (zB−zA)²
As seen from the coordinate transformation above,
xB−xA = [(XB(T)−vX·T) −
− (XA(T)−vX·T)] =
= XB(T)−XA(T)
Analogously,
yB−yA = YB(T)−YA(T)
zB−zA = ZB(T)−ZA(T)
From this immediately follows
dβ²(A,B) = (xB−xA)² +
+ (yB−yA)² + (zB−zA)² =
= (XB−XA)² +
+ (YB−YA)² + (ZB−ZA)² =
= dα²(A,B)
This means that the distance between two points in Euclidian space, as measured in two different inertial reference frame, is the same.
In other words, the Euclidian metrics are invariant relative to Galilean transformation of coordinates.
Consequently, a circle in one inertial frame will look like a circle of the same radius in another, a square in one inertial frame will look exactly the same in another etc.
Newton's Law
of Universal Gravitation
The law of universal gravity between two objects states that there is a force of attraction F between any two objects that is equal to
F = G·m1·m2 /r²
where
F is the force of attraction,
G is the gravitational constant,
m1 is the mass of object 1,
m2 is the mass of object 2,
r is the distance between centers of the masses of these objects.
Observed from two different inertial reference frames, objects are the same, their masses are the same and, as proven above, the distance between them is the same as well.
Therefore, the force of gravity between them is also the same in both reference frames and we can say that the Law of Universal Gravitation is invariant relative to the Galilean transformation.
Coulomb's Law
of Electrostatic Force
The Coulomb's law of electrostatic attracting or repelling force between two electrically charged objects states that there is a force F between any two charged objects that is equal to
F = ke·q1·q2 /r²
where
F is the electrostatic force,
ke is the Coulomb constant,
q1 is the electric charge of object 1,
q2 is the electric charge of object 2,
r is the distance between centers of the charges in these objects.
Observed from two different inertial reference frames, objects are the same, their electric charges are the same and, as proven above, the distance between them is the same as well.
Therefore, the electrostatic force between them is also the same in both reference frames and we can say that the Coulomb's Law of Electrostatic Force between two electrically charged objects is invariant relative to the Galilean transformation.
Thursday, May 25, 2023
Monday, May 22, 2023
Galilean Invariance: UNIZOR.COM - Relativity 4 All - Galilean
Notes to a video lecture on http://www.unizor.com
Galilean Invariance
We stated previously that all laws of Physics are supposed to be expressed in the same form in all inertial reference frames (Principle of Relativity or Galilean Invariance).
This statement we take as an axiom because it corresponds to our experience and agrees with physical experiments.
We have also suggested the Galilean transformations of coordinates as a tool to express the known laws of Physics in different inertial reference frames to verify the invariance of these laws.
Let's check some laws of Mechanics comparing them in two inertial reference frames - α{T,X,Y,Z} and β{t,x,y,z} moving relative to α with constant speed v along the X-axis of α, maintaining parallelism of corresponding axes.
Assume, at zero-time T=t=0 both reference frames coincide and, therefore, any point in space at zero-time has the same coordinates in both reference frames.
We will check if some familiar physical laws look the same in these two inertial systems and are invariant relative to Galilean transformation of coordinates.
Newton's First Law
The Newton's First Law of Motion (Law of Inertia) states that every object in absence of force acting on it will remain at rest or in uniform motion in a straight line.
Assume, an object in α{T,X,Y,Z} frame is moving with a constant speed
V={VX,VY,VZ}
along a straight line.
Its position then can be described as linear functions of time
X(T) = X0 + VX·T
Y(T) = Y0 + VY·T
Z(T) = Z0 + VZ·T
where {X0,Y0,Z0} are coordinates of an initial position of an object at zero-time (t=T=0) in both reference frames and {VX,VY,VZ} are constant components of an object's velocity vector - projections of this vector on three space axes in α{T,X,Y,Z} system that are equal to corresponding projections on coordinate axes in β{t,x,y,z} system because of parallelism of corresponding axes.
Let's apply the transformation to β{t,x,y,z} frame moving along the X-axis of α frame with constant speed v.
t = T
x(t) = X(T) − v·T =
= X0 + (VX − v)·T =
= X0 + (VX − v)·t
y(t) = Y(T) = Y0 + VY·T =
= Y0 + VY·t
z(t) = Z(T) = Z0 + VZ·T =
= Z0 + VZ·t
Thus, the motion in β frame described by these linear functions of time t
x(t) = X0 + (VX − v)·t
y(t) = Y0 + VY·t
z(t) = Z0 + VZ·t
is indeed a uniform motion along a straight line.
The Newton's First Law is preserved by a transformation from one inertial reference frame to another.
Newton's Second Law
The Newton's Second Law of Motion establishes the relationship between the force F, mass m and acceleration a of an object
F = m·a
where F and a are vectors, while m is a positive constant.
That means that acceleration vector is directed along the same direction as a vector of force.
This vector equation in α{T,X,Y,Z} reference frame can be rewritten in coordinate form
FX = m·aX
FY = m·aY
FZ = m·aZ
where {FX,FY,FZ} are constant components of a force vector - projections of this vector on three space axes in α{T,X,Y,Z} system that are equal to corresponding projections on coordinate axes in β{t,x,y,z} system because of parallelism of corresponding axes and
{aX,aY,aZ} are constant components of an acceleration vector, also the same in both systems for the same reason.
As we know, an acceleration is the second derivative of a position (a function of time) by time.
Therefore, in α frame the above equations can be written as
FX = m·X"(T)
FY = m·Y"(T)
FZ = m·Z"(T)
Since coordinates in β
{t,x,y,z} frame are related to coordinates in α frame as
t = T
x(t) = X(T) − v·T
y(t) = Y(T)
z(t) = Z(T)
and v is a constant speed of β{t,x,y,z} reference frame relative to α{T,X,Y,Z},
the first derivative by time from coordinates is
x'(t) = X'(T) − v
y'(t) = Y'(T)
z'(t) = Z'(T)
and the second derivative is
x"(t) = X"(T)
y"(t) = Y"(T)
z"(t) = Z"(T)
As we see, the acceleration of an object is the same in both inertial frames.
As mentioned before, it's important to notice that the components of the force and acceleration vectors are, correspondingly, the same in both reference frames since the axes of coordinates are correspondingly parallel.
Therefore,
Fx = FX = m·X"(T) = m·x"(t)
Fy = FY = m·Y"(T) = m·y"(t)
Fz = FZ = m·Z"(T) = m·z"(t)
That determines the equations of motion in β frame.
Fx = m·x"(t) = m·ax
Fy = m·y"(t) = m·ay
Fz = m·z"(t) = m·az
As we see, equations of motions in both reference frames are identical, which confirms the identical form of the Newton's Second Law if we switch from one inertial frame to another.
Velocity Addition Law
Let's examine the velocity of a moving object in two different inertial reference frames, α{T,X,Y,Z} and β{t,x,y,z}, assuming β frame is moving along X-axis of the frame α with constant speed v along X-axis.
The coordinates of this object in these reference frames are related by a familiar relations
t = T
x(t) = X(T) − v·T
y(t) = Y(T)
z(t) = Z(T)
Consider an object moving in α frame with a constant vector velocity Vα(T).
The coordinates of a velocity vector are first derivatives of coordinates of the position. So, if our object moves in the α{T,X,Y,Z} frame with velocity vector
Vα(T) =
= {VXα(T),VYα(T),VZα(T)}
The components of this vector are, correspondingly
VXα(T) = X'(T)
VYα(T) = Y'(T)
VZα(T) = Z'(T)
Viewed from the β{t,x,y,z} reference frame, its velocity is
Vβ(t) = {Vxβ(t),Vyβ(t),Vzβ(t)}
and the components of this vector are
Vxβ(t) = x'(t)
Vyβ(t) = y'(t)
Vzβ(t) = z'(t)
The equations of coordinate transformation
t = T
x(t) = X(T) − v·T
y(t) = Y(T)
z(t) = Z(T)
applied to components of velocity vector produce
Vxβ(t) = x'(t) = X'(T) − v
Vyβ(t) = y'(t) = Y'(T)
Vzβ(t) = z'(t) = Z'(T)
In vector form the movement of the β reference frame along X-axis of α with speed v means moving along a vector vα(v,0,0)
Therefore, we can write in vector form the relationship between expressions of a movement of an object in two reference frames as
Vβ(t) = Vα(T) − vα
where Vβ(t) is the velocity vector in β reference frame,
Vα(T) is the velocity vector in α reference frame,
vα is a vector describing the motion of β reference frame along X-axis of α.
It's easy to generalize this relationship to a case when reference frame β moves along any constant vector vα(vX,vY,vZ) within frame α, not necessarily along its X-axis.
The equations of coordinate transformation in this more general case are
t = T
x(t) = X(T) − vX·T
y(t) = Y(T) − vY·T
z(t) = Z(T) − vZ·T
applied to components of velocity vector produce
Vxβ(t) = x'(t) = X'(T) − vX
Vyβ(t) = y'(t) = Y'(T) − vY
Vzβ(t) = z'(t) = Z'(T) − vZ
In vector form this relationship between expressions of a movement of an object in two reference frames looks exactly as above:
Vβ(t) = Vα(T) − vα
The above is the velocity addition law of Galilean transformation.
Galilean Invariance
We stated previously that all laws of Physics are supposed to be expressed in the same form in all inertial reference frames (Principle of Relativity or Galilean Invariance).
This statement we take as an axiom because it corresponds to our experience and agrees with physical experiments.
We have also suggested the Galilean transformations of coordinates as a tool to express the known laws of Physics in different inertial reference frames to verify the invariance of these laws.
Let's check some laws of Mechanics comparing them in two inertial reference frames - α{T,X,Y,Z} and β{t,x,y,z} moving relative to α with constant speed v along the X-axis of α, maintaining parallelism of corresponding axes.
Assume, at zero-time T=t=0 both reference frames coincide and, therefore, any point in space at zero-time has the same coordinates in both reference frames.
We will check if some familiar physical laws look the same in these two inertial systems and are invariant relative to Galilean transformation of coordinates.
Newton's First Law
The Newton's First Law of Motion (Law of Inertia) states that every object in absence of force acting on it will remain at rest or in uniform motion in a straight line.
Assume, an object in α{T,X,Y,Z} frame is moving with a constant speed
V={VX,VY,VZ}
along a straight line.
Its position then can be described as linear functions of time
X(T) = X0 + VX·T
Y(T) = Y0 + VY·T
Z(T) = Z0 + VZ·T
where {X0,Y0,Z0} are coordinates of an initial position of an object at zero-time (t=T=0) in both reference frames and {VX,VY,VZ} are constant components of an object's velocity vector - projections of this vector on three space axes in α{T,X,Y,Z} system that are equal to corresponding projections on coordinate axes in β{t,x,y,z} system because of parallelism of corresponding axes.
Let's apply the transformation to β{t,x,y,z} frame moving along the X-axis of α frame with constant speed v.
t = T
x(t) = X(T) − v·T =
= X0 + (VX − v)·T =
= X0 + (VX − v)·t
y(t) = Y(T) = Y0 + VY·T =
= Y0 + VY·t
z(t) = Z(T) = Z0 + VZ·T =
= Z0 + VZ·t
Thus, the motion in β frame described by these linear functions of time t
x(t) = X0 + (VX − v)·t
y(t) = Y0 + VY·t
z(t) = Z0 + VZ·t
is indeed a uniform motion along a straight line.
The Newton's First Law is preserved by a transformation from one inertial reference frame to another.
Newton's Second Law
The Newton's Second Law of Motion establishes the relationship between the force F, mass m and acceleration a of an object
F = m·a
where F and a are vectors, while m is a positive constant.
That means that acceleration vector is directed along the same direction as a vector of force.
This vector equation in α{T,X,Y,Z} reference frame can be rewritten in coordinate form
FX = m·aX
FY = m·aY
FZ = m·aZ
where {FX,FY,FZ} are constant components of a force vector - projections of this vector on three space axes in α{T,X,Y,Z} system that are equal to corresponding projections on coordinate axes in β{t,x,y,z} system because of parallelism of corresponding axes and
{aX,aY,aZ} are constant components of an acceleration vector, also the same in both systems for the same reason.
As we know, an acceleration is the second derivative of a position (a function of time) by time.
Therefore, in α frame the above equations can be written as
FX = m·X"(T)
FY = m·Y"(T)
FZ = m·Z"(T)
Since coordinates in β
{t,x,y,z} frame are related to coordinates in α frame as
t = T
x(t) = X(T) − v·T
y(t) = Y(T)
z(t) = Z(T)
and v is a constant speed of β{t,x,y,z} reference frame relative to α{T,X,Y,Z},
the first derivative by time from coordinates is
x'(t) = X'(T) − v
y'(t) = Y'(T)
z'(t) = Z'(T)
and the second derivative is
x"(t) = X"(T)
y"(t) = Y"(T)
z"(t) = Z"(T)
As we see, the acceleration of an object is the same in both inertial frames.
As mentioned before, it's important to notice that the components of the force and acceleration vectors are, correspondingly, the same in both reference frames since the axes of coordinates are correspondingly parallel.
Therefore,
Fx = FX = m·X"(T) = m·x"(t)
Fy = FY = m·Y"(T) = m·y"(t)
Fz = FZ = m·Z"(T) = m·z"(t)
That determines the equations of motion in β frame.
Fx = m·x"(t) = m·ax
Fy = m·y"(t) = m·ay
Fz = m·z"(t) = m·az
As we see, equations of motions in both reference frames are identical, which confirms the identical form of the Newton's Second Law if we switch from one inertial frame to another.
Velocity Addition Law
Let's examine the velocity of a moving object in two different inertial reference frames, α{T,X,Y,Z} and β{t,x,y,z}, assuming β frame is moving along X-axis of the frame α with constant speed v along X-axis.
The coordinates of this object in these reference frames are related by a familiar relations
t = T
x(t) = X(T) − v·T
y(t) = Y(T)
z(t) = Z(T)
Consider an object moving in α frame with a constant vector velocity Vα(T).
The coordinates of a velocity vector are first derivatives of coordinates of the position. So, if our object moves in the α{T,X,Y,Z} frame with velocity vector
Vα(T) =
= {VXα(T),VYα(T),VZα(T)}
The components of this vector are, correspondingly
VXα(T) = X'(T)
VYα(T) = Y'(T)
VZα(T) = Z'(T)
Viewed from the β{t,x,y,z} reference frame, its velocity is
Vβ(t) = {Vxβ(t),Vyβ(t),Vzβ(t)}
and the components of this vector are
Vxβ(t) = x'(t)
Vyβ(t) = y'(t)
Vzβ(t) = z'(t)
The equations of coordinate transformation
t = T
x(t) = X(T) − v·T
y(t) = Y(T)
z(t) = Z(T)
applied to components of velocity vector produce
Vxβ(t) = x'(t) = X'(T) − v
Vyβ(t) = y'(t) = Y'(T)
Vzβ(t) = z'(t) = Z'(T)
In vector form the movement of the β reference frame along X-axis of α with speed v means moving along a vector vα(v,0,0)
Therefore, we can write in vector form the relationship between expressions of a movement of an object in two reference frames as
Vβ(t) = Vα(T) − vα
where Vβ(t) is the velocity vector in β reference frame,
Vα(T) is the velocity vector in α reference frame,
vα is a vector describing the motion of β reference frame along X-axis of α.
It's easy to generalize this relationship to a case when reference frame β moves along any constant vector vα(vX,vY,vZ) within frame α, not necessarily along its X-axis.
The equations of coordinate transformation in this more general case are
t = T
x(t) = X(T) − vX·T
y(t) = Y(T) − vY·T
z(t) = Z(T) − vZ·T
applied to components of velocity vector produce
Vxβ(t) = x'(t) = X'(T) − vX
Vyβ(t) = y'(t) = Y'(T) − vY
Vzβ(t) = z'(t) = Z'(T) − vZ
In vector form this relationship between expressions of a movement of an object in two reference frames looks exactly as above:
Vβ(t) = Vα(T) − vα
The above is the velocity addition law of Galilean transformation.
Tuesday, May 16, 2023
Galilean Transformation: UNIZOR.COM - Relativity 4 All - Galilean
Notes to a video lecture on http://www.unizor.com
Galilean Transformations
Here and in many other places of this course we will be dealing with two inertial reference frames moving relative to each other with constant speed.
Let's call one of these reference frames primary or α and its Cartesian axes will be called X,Y,Z.
The other system will be called secondary or β with axes x,y,z.
For simplicity, we assume that β reference frame at zero-time t=0 coincides with α.
We also assume that β frame is moving along the X-axis of α with constant speed V, preserving the parallelism of corresponding axes of these reference frames.
Time is considered absolute and the same for both reference frames.
Assume, there is a given stationary point P with coordinates {XP,YP,ZP} not changing with time in α frame.
Our task is to determine its coordinates {xP(t),yP(t),zP(t)} in β frame that moves relative to α as described above.
It is geometrically obvious that coordinates YP and ZP of this point in α will be equal to coordinates yP(t) and zP(t) of this point in β.
Coordinate xP(t) of this point in β will be shifted in time from its coordinate XP in α. This shift will be zero at zero-time t=0 and, as time goes by, it will shift more and more proportional to speed V of the movement of β relative to α.
More exactly, since the origin of β moves relative to the origin of α with speed V along X-axis, the X-coordinate of β's origin in α at time t is Xβ(t)=V·t.
Therefore, since X-coordinate of point P is constant XP, its x-coordinate in β frame is xP(t)=XP−V·t.
We can summarize the transformation of coordinates between primary and secondary frame as follows
xP(t) = XP−V·t
yP(t) = YP
zP(t) = ZP
If point P is not a stationary point in α frame, but is moving, and its coordinates are functions of absolute time {XP(t),YP(t),ZP(t)}, the transformation to β frame would be analogous
xP(t) = XP(t)−V·t
yP(t) = YP(t)
zP(t) = ZP(t)
Sometimes these transformations are complemented with t=T, implying that T is time in α frame, while t is time in β frame, to emphasize that the time is absolute and the same in both reference frames.
Since P is any fixed point in α frame, there is no need to index these formulae of transformation and they can be written as
t = T
x(t) = X(T)−V·T
y(t) = Y(T)
z(t) = Z(T)
These transformations of coordinates from one inertial reference frame to another, moving relative to the first as described above, are called
Galilean Transformations.
Let's consider the same relative movement of the same two reference frames α and β from the point of view of an observer at rest in the β frame.
From his perspective its own β frame is stationary and α frame is moving along x-axis with speed −V.
All the considerations above can now be applied to this situation and the transformation of coordinates from β to α will be similar to the above, but with a reversed speed of movement of α frame relative to β:
T = t
X(T) = x(t)+V·t
Y(T) = y(t)
Z(T) = z(t)
Linear transformation of coordinates can be expressed in matrix and vectors form.
We can express the transformation from α frame to β frame as
The transformation from β frame to α frame is similar, but the speed is reversed
Just to check if two above transformations are inverse to each other, we can multiply the transformation matrices according to the rule of multiplication of matrices and will get an identity matrix 4x4 with all diagonal elements equal to 1 and all other elements equal to 0, as supposed to be with inverse matrices.
Incidentally, the determinant of both transformation matrices equals to 1.
Galilean Transformations
Here and in many other places of this course we will be dealing with two inertial reference frames moving relative to each other with constant speed.
Let's call one of these reference frames primary or α and its Cartesian axes will be called X,Y,Z.
The other system will be called secondary or β with axes x,y,z.
For simplicity, we assume that β reference frame at zero-time t=0 coincides with α.
We also assume that β frame is moving along the X-axis of α with constant speed V, preserving the parallelism of corresponding axes of these reference frames.
Time is considered absolute and the same for both reference frames.
Assume, there is a given stationary point P with coordinates {XP,YP,ZP} not changing with time in α frame.
Our task is to determine its coordinates {xP(t),yP(t),zP(t)} in β frame that moves relative to α as described above.
It is geometrically obvious that coordinates YP and ZP of this point in α will be equal to coordinates yP(t) and zP(t) of this point in β.
Coordinate xP(t) of this point in β will be shifted in time from its coordinate XP in α. This shift will be zero at zero-time t=0 and, as time goes by, it will shift more and more proportional to speed V of the movement of β relative to α.
More exactly, since the origin of β moves relative to the origin of α with speed V along X-axis, the X-coordinate of β's origin in α at time t is Xβ(t)=V·t.
Therefore, since X-coordinate of point P is constant XP, its x-coordinate in β frame is xP(t)=XP−V·t.
We can summarize the transformation of coordinates between primary and secondary frame as follows
xP(t) = XP−V·t
yP(t) = YP
zP(t) = ZP
If point P is not a stationary point in α frame, but is moving, and its coordinates are functions of absolute time {XP(t),YP(t),ZP(t)}, the transformation to β frame would be analogous
xP(t) = XP(t)−V·t
yP(t) = YP(t)
zP(t) = ZP(t)
Sometimes these transformations are complemented with t=T, implying that T is time in α frame, while t is time in β frame, to emphasize that the time is absolute and the same in both reference frames.
Since P is any fixed point in α frame, there is no need to index these formulae of transformation and they can be written as
t = T
x(t) = X(T)−V·T
y(t) = Y(T)
z(t) = Z(T)
These transformations of coordinates from one inertial reference frame to another, moving relative to the first as described above, are called
Galilean Transformations.
Let's consider the same relative movement of the same two reference frames α and β from the point of view of an observer at rest in the β frame.
From his perspective its own β frame is stationary and α frame is moving along x-axis with speed −V.
All the considerations above can now be applied to this situation and the transformation of coordinates from β to α will be similar to the above, but with a reversed speed of movement of α frame relative to β:
T = t
X(T) = x(t)+V·t
Y(T) = y(t)
Z(T) = z(t)
Linear transformation of coordinates can be expressed in matrix and vectors form.
We can express the transformation from α frame to β frame as
| t |
| x |
| y |
| z |
| = |
| 1 | 0 | 0 | 0 |
| −V | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| · |
| T |
| X |
| Y |
| Z |
The transformation from β frame to α frame is similar, but the speed is reversed
| T |
| X |
| Y |
| Z |
| = |
| 1 | 0 | 0 | 0 |
| V | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| · |
| t |
| x |
| y |
| z |
Just to check if two above transformations are inverse to each other, we can multiply the transformation matrices according to the rule of multiplication of matrices and will get an identity matrix 4x4 with all diagonal elements equal to 1 and all other elements equal to 0, as supposed to be with inverse matrices.
Incidentally, the determinant of both transformation matrices equals to 1.
Reference Frames and Principle of Relativity: UNIZOR.COM - Relativity4All - Galilean
Notes to a video lecture on http://www.unizor.com
Reference Frames and
Principle of Relativity
In Physics we use numbers to describe the motion of objects, like distance, speed, time period etc.
These numbers are derived from certain system of coordinates in space to measure space-related characteristics and some devices to measure time, like clocks.
In this course we will use three-dimensional Cartesian system of space coordinates and some time origin to measure time intervals.
We will not choose any particular Cartesian system of coordinates with fixed origin and directions of axes (like originated in the center of the Sun and the X-axis directed toward a star Sirius).
Instead, we assume the existence of some system of Cartesian coordinates where any object with zero cumulative force acting upon it is moving along a straight line with a constant speed (speed might be zero, in which case the trajectory is not a straight line but a point).
We will call such a system of coordinates an inertial reference frame and will refer to it as "primary".
Obviously, there is no absolutely inertial system with absolutely no external forces, this is an abstraction. But many systems, including the one mentioned above related to stars are as good an approximation to an inertial reference frame as it can be.
Notice that any other Cartesian system of coordinates that moves in such a way that its origin moves in the "primary" system along a straight line with constant linear speed, and its axes are always parallel to axes of the "primary" system would also qualify for being called inertial reference frame.
Also, we will not choose any particular moment in time as the beginning of time (like midnight on January 1st, 1900). Any moment will do, as long as it's fixed and called the "beginning of time" or "zero-time".
With an introduction of inertial reference frames we can state an important principle of classical Physics - all the Physics laws must be the same if expressed quantitatively, using corresponding coordinates and time, in all inertial reference frames.
This principle is called Galilean Invariance or Galilean Relativity Principle and, in other words, it states that identical experiments conducted within two different inertial reference frames, that move relative to each other, produce identical results.
The parallel (no rotation) and uniform (regarding velocity) movement of inertial reference frame α relative to another one β (uniform translation of α relative to β) cannot be experimentally detected from inside frame α or from inside frame β.
It needs an external observer, who sees both α and β moving relative to each other, to come up with quantitative judgement about this relative motion.
Consequently, there is no system of coordinates that can be considered as absolutely "at rest", while others are moving relative to it. For any reference frame α there is another β moving relatively to it, while, for the same token, frame α can be considered as moving relatively to frame β.
This is the essence of Principle of Relativity.
Applied to inertial reference frames, this principle can be formulated as all laws of Physics are expressed in the same form in all inertial reference frames.
This is called the Principle of Relativity.
Reference Frames and
Principle of Relativity
In Physics we use numbers to describe the motion of objects, like distance, speed, time period etc.
These numbers are derived from certain system of coordinates in space to measure space-related characteristics and some devices to measure time, like clocks.
In this course we will use three-dimensional Cartesian system of space coordinates and some time origin to measure time intervals.
We will not choose any particular Cartesian system of coordinates with fixed origin and directions of axes (like originated in the center of the Sun and the X-axis directed toward a star Sirius).
Instead, we assume the existence of some system of Cartesian coordinates where any object with zero cumulative force acting upon it is moving along a straight line with a constant speed (speed might be zero, in which case the trajectory is not a straight line but a point).
We will call such a system of coordinates an inertial reference frame and will refer to it as "primary".
Obviously, there is no absolutely inertial system with absolutely no external forces, this is an abstraction. But many systems, including the one mentioned above related to stars are as good an approximation to an inertial reference frame as it can be.
Notice that any other Cartesian system of coordinates that moves in such a way that its origin moves in the "primary" system along a straight line with constant linear speed, and its axes are always parallel to axes of the "primary" system would also qualify for being called inertial reference frame.
Also, we will not choose any particular moment in time as the beginning of time (like midnight on January 1st, 1900). Any moment will do, as long as it's fixed and called the "beginning of time" or "zero-time".
With an introduction of inertial reference frames we can state an important principle of classical Physics - all the Physics laws must be the same if expressed quantitatively, using corresponding coordinates and time, in all inertial reference frames.
This principle is called Galilean Invariance or Galilean Relativity Principle and, in other words, it states that identical experiments conducted within two different inertial reference frames, that move relative to each other, produce identical results.
The parallel (no rotation) and uniform (regarding velocity) movement of inertial reference frame α relative to another one β (uniform translation of α relative to β) cannot be experimentally detected from inside frame α or from inside frame β.
It needs an external observer, who sees both α and β moving relative to each other, to come up with quantitative judgement about this relative motion.
Consequently, there is no system of coordinates that can be considered as absolutely "at rest", while others are moving relative to it. For any reference frame α there is another β moving relatively to it, while, for the same token, frame α can be considered as moving relatively to frame β.
This is the essence of Principle of Relativity.
Applied to inertial reference frames, this principle can be formulated as all laws of Physics are expressed in the same form in all inertial reference frames.
This is called the Principle of Relativity.
Introduction to Special Theory of Relativity: UNIZOR.COM - Relativity4All
Notes to a video lecture on http://www.unizor.com
Introduction to
Special Theory of Relativity
This course presented on UNIZOR.COM is called Relativity 4 All because its purpose is to introduce the concepts of the Special Theory of Relativity on a sufficiently rigorous mathematical basis, yet in as plain a language as possible to be understood by a non-physicist, who is just curious about what exactly Albert Einstein has come up with in the beginning of the 20th century and why his concepts revolutionized the contemporary Physics.
At the same time, certain level of knowledge of Math and Physics is required to understand Special Theory of Relativity.
An adequate amount of knowledge can be obtained from this site UNIZOR.COM in courses Math 4 Teens and Physics 4 Teens.
Introduction to
Special Theory of Relativity
This course presented on UNIZOR.COM is called Relativity 4 All because its purpose is to introduce the concepts of the Special Theory of Relativity on a sufficiently rigorous mathematical basis, yet in as plain a language as possible to be understood by a non-physicist, who is just curious about what exactly Albert Einstein has come up with in the beginning of the 20th century and why his concepts revolutionized the contemporary Physics.
At the same time, certain level of knowledge of Math and Physics is required to understand Special Theory of Relativity.
An adequate amount of knowledge can be obtained from this site UNIZOR.COM in courses Math 4 Teens and Physics 4 Teens.
Saturday, May 6, 2023
Derived Waves SI Units: UNIZOR.COM - Physics4Teens - Units in Physics - ...
Notes to a video lecture on http://www.unizor.com
For a short introduction to the International System of units (SI) see the previous lecture SI Intro & Time within topic Base SI Units.
Waves Units
All SI units used in the science of Waves are derived from base units introduced in the previous lectures of this course.
Here we will address only uniform transverse waves. Longitudinal waves have analogous units of measurement.
Frequency
The wave frequency is the number of oscillations per second.
Therefore, the unit of measurement for a frequency is 1 oscillation per second (1/s) called hertz (Hz).
1 Hz = 1 1/s
means the wave makes one oscillation per second.
Period
The wave period is an amount of time it takes for a wave to complete one oscillation.
The unit of time and, therefore, of a period is
1 second (s).
Wave Length
The wave length is a distance between two consecutive peaks of the wave.
The unit of distance and, therefore, of wave length is
1 meter (m).
Wave Speed
For a wave propagating in space the wave speed is a speed of any particular peak of the wave.
The unit of speed and, therefore, of wave speed is
1 meter per second (m/s).
For a short introduction to the International System of units (SI) see the previous lecture SI Intro & Time within topic Base SI Units.
Waves Units
All SI units used in the science of Waves are derived from base units introduced in the previous lectures of this course.
Here we will address only uniform transverse waves. Longitudinal waves have analogous units of measurement.
Frequency
The wave frequency is the number of oscillations per second.
Therefore, the unit of measurement for a frequency is 1 oscillation per second (1/s) called hertz (Hz).
1 Hz = 1 1/s
means the wave makes one oscillation per second.
Period
The wave period is an amount of time it takes for a wave to complete one oscillation.
The unit of time and, therefore, of a period is
1 second (s).
Wave Length
The wave length is a distance between two consecutive peaks of the wave.
The unit of distance and, therefore, of wave length is
1 meter (m).
Wave Speed
For a wave propagating in space the wave speed is a speed of any particular peak of the wave.
The unit of speed and, therefore, of wave speed is
1 meter per second (m/s).
Derived Atoms SI Units: UNIZOR.COM - Physics4Teens - Units in Physics - ...
Notes to a video lecture on http://www.unizor.com
For a short introduction to the International System of units (SI) see the previous lecture SI Intro & Time within topic Base SI Units.
Atoms Units
All SI units used in the science of Atoms are derived from base units introduced in the previous lectures of this course.
Unified Atomic Mass Unit
The unified atomic mass unit (abbreviated as u or AMU, or Da in honor of John Dalton, who suggested this unit) is defined as 1/12th of the mass of the atom of carbon-12.
From this definition immediately follows that unified atomic mass unit of carbon-12 is 12u.
To express this unit in kilograms, the units of mass in SI, we can use the Avogadro constant that tells us that 1 mole of carbon-12 (12 gram) contains EXACTLY 6.02214076·1023 atoms.
Therefore, one atom of carbon-12 has a mass of 12/(6.02214076·1023) gram and 1/12th of an atom of carbon-12 has a mass of 1/(6.02214076·1023) gram, which is, approximately, 1.6605391·10−24 gram, from which follows
1 u ~= 1.6605391·10−27 kg
Since 1u is 1/12th of the mass of the atom of carbon-12 and this atom has 6 protons and 6 neutrons in its nucleus plus 6 much smaller electrons, 1u is, approximate, an average mass between a single proton and a single neutron.
Therefore, measured in unified atomic mass units, the atomic mass of any element approximately equals to the number of protons and neutrons in its nucleus.
Electron-Volt
The unit electron-volt (eV) is a measure of kinetic energy an electron gains when it moves from one point of the electrostatic field to another with a difference of potential1 volt (V) between these points.
To express this amount of energy in SI units joules, recall that the difference of potential1 volt (V) between two points means that it requires 1 joule of energy to move 1 coulomb electric charge between these points.
Therefore, reducing electric charge from 1 coulomb (C) to the amount of charge of one electron (e), we will reduce amount of work needed in joules by the same factor.
We know the charge of an electron in SI is EXACTLY
1 e = 1.602176634·10−19 C
from which follows that
1 eV = 1.602176634·10−19 J
For a short introduction to the International System of units (SI) see the previous lecture SI Intro & Time within topic Base SI Units.
Atoms Units
All SI units used in the science of Atoms are derived from base units introduced in the previous lectures of this course.
Unified Atomic Mass Unit
The unified atomic mass unit (abbreviated as u or AMU, or Da in honor of John Dalton, who suggested this unit) is defined as 1/12th of the mass of the atom of carbon-12.
From this definition immediately follows that unified atomic mass unit of carbon-12 is 12u.
To express this unit in kilograms, the units of mass in SI, we can use the Avogadro constant that tells us that 1 mole of carbon-12 (12 gram) contains EXACTLY 6.02214076·1023 atoms.
Therefore, one atom of carbon-12 has a mass of 12/(6.02214076·1023) gram and 1/12th of an atom of carbon-12 has a mass of 1/(6.02214076·1023) gram, which is, approximately, 1.6605391·10−24 gram, from which follows
1 u ~= 1.6605391·10−27 kg
Since 1u is 1/12th of the mass of the atom of carbon-12 and this atom has 6 protons and 6 neutrons in its nucleus plus 6 much smaller electrons, 1u is, approximate, an average mass between a single proton and a single neutron.
Therefore, measured in unified atomic mass units, the atomic mass of any element approximately equals to the number of protons and neutrons in its nucleus.
Electron-Volt
The unit electron-volt (eV) is a measure of kinetic energy an electron gains when it moves from one point of the electrostatic field to another with a difference of potential
To express this amount of energy in SI units joules, recall that the difference of potential
Therefore, reducing electric charge from 1 coulomb (C) to the amount of charge of one electron (e), we will reduce amount of work needed in joules by the same factor.
We know the charge of an electron in SI is EXACTLY
1 e = 1.602176634·10−19 C
from which follows that
1 eV = 1.602176634·10−19 J
Thursday, May 4, 2023
Derived Magnetism SI Units: UNIZOR.COM - Physics4Teens - Units in Physic...
Notes to a video lecture on http://www.unizor.com
For a short introduction to the International System of units (SI) see the previous lecture SI Intro & Time within topic Base SI Units.
Magnetism Units
All SI units used in the science of Magnetism are derived from base units introduced in the previous chapters of this part Units in Physics of the course.
Intensity (B)
Recall the Lorentz force that magnetic field exhorts on a conductor with an electric current flowing in it.
In short, when a conductor with an electric current I and length L is positioned perpendicular to uniform magnetic field lines, the force F of the magnetic field onto a conductor is perpendicular to both, magnetic field lines and a conductor and its magnitude is proportional to both the amperage of an electric current and the length of a conductor
F = B·I·L
where coefficient B is called the intensity or magnetic flux density of the magnetic field.
The above equation allows to establish a unit of measurement of the intensity of the magnetic field as follows.
A uniform magnetic field that exerts a strength of 1 newton (N) on a thin perpendicular to field lines wire of 1 meter (m) long with an electric current of 1 ampere (A) running through this wire has a strength of 1 tesla (T).
Tesla is a unit of measurement of the magnitude of the intensity of a magnetic field.
1 T = 1 N/(A·m)
Its direction is defined as being tangential and co-directional to magnetic field lines.
Magnetic Flux
The strength of a magnetic field is defined by its intensity (or magnetic flux density) at each point of space where the field is present regardless of whether there is a probing device (a conductor with an electric current running through it) or not.
In a sense, it's similar to a concept of pressure of air that exists at any point where air is present regardless of the presence of a manometer that measures this pressure. It represents a force that molecules in their chaotic motion exert onto a unit of area.
While magnetic flux density can be compare to a force that molecules exert onto a unit of area (air pressure), magnetic flux is comparable to a total pressure on any area, which is a product of a pressure onto a unit of area by a size of an area.
So, magnetic flux Φ is a product of intensity B (magnetic flux density) of a magnetic field by an area A perpendicular to magnetic field lines that coincide with a direction of vectors of intensity of the magnetic field
Φ = B·A
From the formula above, the unit of magnetic flux is the magnetic flux of a field of intensity of 1 tesla (T) with its lines going through an area of 1 m² positioned perpendicularly to magnetic field lines.
This unit is called weber (Wb)
1 Wb = 1 T·m²
For a short introduction to the International System of units (SI) see the previous lecture SI Intro & Time within topic Base SI Units.
Magnetism Units
All SI units used in the science of Magnetism are derived from base units introduced in the previous chapters of this part Units in Physics of the course.
Intensity (B)
Recall the Lorentz force that magnetic field exhorts on a conductor with an electric current flowing in it.
In short, when a conductor with an electric current I and length L is positioned perpendicular to uniform magnetic field lines, the force F of the magnetic field onto a conductor is perpendicular to both, magnetic field lines and a conductor and its magnitude is proportional to both the amperage of an electric current and the length of a conductor
F = B·I·L
where coefficient B is called the intensity or magnetic flux density of the magnetic field.
The above equation allows to establish a unit of measurement of the intensity of the magnetic field as follows.
A uniform magnetic field that exerts a strength of 1 newton (N) on a thin perpendicular to field lines wire of 1 meter (m) long with an electric current of 1 ampere (A) running through this wire has a strength of 1 tesla (T).
Tesla is a unit of measurement of the magnitude of the intensity of a magnetic field.
1 T = 1 N/(A·m)
Its direction is defined as being tangential and co-directional to magnetic field lines.
Magnetic Flux
The strength of a magnetic field is defined by its intensity (or magnetic flux density) at each point of space where the field is present regardless of whether there is a probing device (a conductor with an electric current running through it) or not.
In a sense, it's similar to a concept of pressure of air that exists at any point where air is present regardless of the presence of a manometer that measures this pressure. It represents a force that molecules in their chaotic motion exert onto a unit of area.
While magnetic flux density can be compare to a force that molecules exert onto a unit of area (air pressure), magnetic flux is comparable to a total pressure on any area, which is a product of a pressure onto a unit of area by a size of an area.
So, magnetic flux Φ is a product of intensity B (magnetic flux density) of a magnetic field by an area A perpendicular to magnetic field lines that coincide with a direction of vectors of intensity of the magnetic field
Φ = B·A
From the formula above, the unit of magnetic flux is the magnetic flux of a field of intensity of 1 tesla (T) with its lines going through an area of 1 m² positioned perpendicularly to magnetic field lines.
This unit is called weber (Wb)
1 Wb = 1 T·m²
Wednesday, May 3, 2023
Derived Electricity SI Units: UNIZOR.COM - Physics4Teens - Units in Phys...
Notes to a video lecture on http://www.unizor.com
For a short introduction to the International System of units (SI) see the previous lecture "SI Intro & Time" within this topic.
Electricity Units
All SI units used in the science of Electricity are derived from base units introduced in the previous chapter Base SI Units of this part Units in Physics of the course.
Electric Charge
As we know, electric charge is excess (negative charge) or deficit (positive charge) of electrons relatively to the number of protons in the atoms of an object.
Electric charge (that is, excess or deficit of electrons) can travel along a conductor forming an electric current.
There is a base SI unit of electric current - ampere (A).
An amount of electrons transferred by a current of 1 ampere during 1 second is a unit of measuring the electric charge, it's called coulomb (C)
1 C = 1 A·s
When electrons are going away from an object by a current of 1A during the time of 1s, this object is positively charged with 1C of electric charge.
When electrons are going into an object by a current of 1A during the time of 1s, this object is negatively charged with 1C of electric charge.
Voltage
Consider any electrostatic field. For example, an electrostatic field around some negatively charged point-object.
Every positively charged point-object that carries a charge of +1 coulomb (test charge) will experience certain force in this field. In our example of a field produced by a negatively charged point-object the force will be an attraction towards the center of a field.
Therefore, to move our test charge from point A to point B of the electrostatic field might require certain energy to spend against the force of a field. In our example of a field produced by a negatively charged point-object, to move a test charge away from a center of the field requires to do some work against the force of attraction.
Obviously, since the force of attraction or repelling between charges is proportional to the charges themselves (the Coulomb's Law), the amount of work to move any charge from point A to point B will be proportional to this charge.
If our test charge of +1 coulomb requires to perform 1 joule of work to move it from point A to point B in the electrostatic field, we say that there is a 1 volt (V) of difference in electric potential between these points A and B.
This difference in electric potential is called voltage.
So, the unit of voltage is 1 volt, which can be expressed as
1 V = 1 J/C
Resistance
Measurement of electric resistance is based on the Ohm's Law. Stronger electric field produced by a greater difference in electric potential (voltage) between the ends of a conductor should cause proportionally more intense movement of electrons, greater number of electrons going from one end of a conductor and reaching the other end per unit of time (amperage)
I = U/R
where
I is electric current,
U is voltage,
R is electric resistance.
If the voltage between two opposite points of a conductor is constant 1 volt (V) and the electric current running through this conductor is 1 ampere (A), we say that a conductor's resistance equals to 1 ohm (Ω).
So, the unit of resistance is 1 ohm, which can be expressed as
1 Ω = 1 V/A
For a short introduction to the International System of units (SI) see the previous lecture "SI Intro & Time" within this topic.
Electricity Units
All SI units used in the science of Electricity are derived from base units introduced in the previous chapter Base SI Units of this part Units in Physics of the course.
Electric Charge
As we know, electric charge is excess (negative charge) or deficit (positive charge) of electrons relatively to the number of protons in the atoms of an object.
Electric charge (that is, excess or deficit of electrons) can travel along a conductor forming an electric current.
There is a base SI unit of electric current - ampere (A).
An amount of electrons transferred by a current of 1 ampere during 1 second is a unit of measuring the electric charge, it's called coulomb (C)
1 C = 1 A·s
When electrons are going away from an object by a current of 1A during the time of 1s, this object is positively charged with 1C of electric charge.
When electrons are going into an object by a current of 1A during the time of 1s, this object is negatively charged with 1C of electric charge.
Voltage
Consider any electrostatic field. For example, an electrostatic field around some negatively charged point-object.
Every positively charged point-object that carries a charge of +1 coulomb (test charge) will experience certain force in this field. In our example of a field produced by a negatively charged point-object the force will be an attraction towards the center of a field.
Therefore, to move our test charge from point A to point B of the electrostatic field might require certain energy to spend against the force of a field. In our example of a field produced by a negatively charged point-object, to move a test charge away from a center of the field requires to do some work against the force of attraction.
Obviously, since the force of attraction or repelling between charges is proportional to the charges themselves (the Coulomb's Law), the amount of work to move any charge from point A to point B will be proportional to this charge.
If our test charge of +1 coulomb requires to perform 1 joule of work to move it from point A to point B in the electrostatic field, we say that there is a 1 volt (V) of difference in electric potential between these points A and B.
This difference in electric potential is called voltage.
So, the unit of voltage is 1 volt, which can be expressed as
1 V = 1 J/C
Resistance
Measurement of electric resistance is based on the Ohm's Law. Stronger electric field produced by a greater difference in electric potential (voltage) between the ends of a conductor should cause proportionally more intense movement of electrons, greater number of electrons going from one end of a conductor and reaching the other end per unit of time (amperage)
I = U/R
where
I is electric current,
U is voltage,
R is electric resistance.
If the voltage between two opposite points of a conductor is constant 1 volt (V) and the electric current running through this conductor is 1 ampere (A), we say that a conductor's resistance equals to 1 ohm (Ω).
So, the unit of resistance is 1 ohm, which can be expressed as
1 Ω = 1 V/A
Derived Energy SI Units: UNIZOR.COM - Physics4Teens - Units in Physics -...
Notes to a video lecture on http://www.unizor.com
For a short introduction to the International System of units (SI) see the previous lecture "SI Intro & Time" within this topic.
Energy Units
All SI units used in the science of Energy are derived from base units introduced in the previous chapter Base SI Units of this part Units in Physics of the course.
Work
In a simple case of a constant force F accelerating an object along a straight line in the direction of the force the amount of work W done by this force can be measured by a product of the magnitude of a force by a distance s it pushed an object
W = F · s
This simple case is the base for introduction of a unit of work joule (J) as an amount of work done by a force of 1 newton (N) on a distance of 1 meter (m)
1 J = 1 N·m
or, using only the base SI units with 1N=1kg·m/s²,
1 J = 1 kg·m²/s²
In a more complicated case of a force acting at an angle to a trajectory we represent a vector of force as a sum of two vectors, one parallel to a trajectory and another perpendicular to it.
The component of a force that goes parallel to a trajectory is the one that does the work, while a perpendicular to it component does not move an object along a trajectory and does not work towards this movement.
So, to calculate the work performed by such a force, we use only its projection Ftraj onto a trajectory that can be expressed in terms of a full magnitude of a force F and an angle φ between the vector of force and direction of a movement
Ftraj = F· cos(φ)
This complication does not change the fact that work can be measured in joules defined above. Work in joules is still a product of a force along a trajectory in newtons by a distance in meters.
Power
Power is an amount of work done in a unit of time.
The unit of measuring power in SI is joules per second. This unit is called watt and one watt (W) is a joule (J) per second (s)
1 W = 1 J/s
or, using only the base SI units with 1J=1kg·m²/s²,
1 W = 1 kg·m²/s³
Energy
Let's talk about mechanical energy here.
When object X exerts a force moving another object Y, changing its position, direction or speed, thus performing some work, we say that object X has spend certain amount of energy by forcing object Y to move.
The amount of energy spent is, by definition, exactly the amount of work performed.
Therefore, an amount of energy can be measured in SI in exactly the same units as an amount of work, that is in joules (J).
In the Energy part of this course we introduced a concept of energy manifested not only in mechanical movement, but also in chemical reactions, heat, electromagnetism, gravitation, nuclear reactions etc.
All these types of energy can still be measured in joules because at the very base level we still deal with some forces and movements caused by them.
For a short introduction to the International System of units (SI) see the previous lecture "SI Intro & Time" within this topic.
Energy Units
All SI units used in the science of Energy are derived from base units introduced in the previous chapter Base SI Units of this part Units in Physics of the course.
Work
In a simple case of a constant force F accelerating an object along a straight line in the direction of the force the amount of work W done by this force can be measured by a product of the magnitude of a force by a distance s it pushed an object
W = F · s
This simple case is the base for introduction of a unit of work joule (J) as an amount of work done by a force of 1 newton (N) on a distance of 1 meter (m)
1 J = 1 N·m
or, using only the base SI units with 1N=1kg·m/s²,
1 J = 1 kg·m²/s²
In a more complicated case of a force acting at an angle to a trajectory we represent a vector of force as a sum of two vectors, one parallel to a trajectory and another perpendicular to it.
The component of a force that goes parallel to a trajectory is the one that does the work, while a perpendicular to it component does not move an object along a trajectory and does not work towards this movement.
So, to calculate the work performed by such a force, we use only its projection Ftraj onto a trajectory that can be expressed in terms of a full magnitude of a force F and an angle φ between the vector of force and direction of a movement
Ftraj = F· cos(φ)
This complication does not change the fact that work can be measured in joules defined above. Work in joules is still a product of a force along a trajectory in newtons by a distance in meters.
Power
Power is an amount of work done in a unit of time.
The unit of measuring power in SI is joules per second. This unit is called watt and one watt (W) is a joule (J) per second (s)
1 W = 1 J/s
or, using only the base SI units with 1J=1kg·m²/s²,
1 W = 1 kg·m²/s³
Energy
Let's talk about mechanical energy here.
When object X exerts a force moving another object Y, changing its position, direction or speed, thus performing some work, we say that object X has spend certain amount of energy by forcing object Y to move.
The amount of energy spent is, by definition, exactly the amount of work performed.
Therefore, an amount of energy can be measured in SI in exactly the same units as an amount of work, that is in joules (J).
In the Energy part of this course we introduced a concept of energy manifested not only in mechanical movement, but also in chemical reactions, heat, electromagnetism, gravitation, nuclear reactions etc.
All these types of energy can still be measured in joules because at the very base level we still deal with some forces and movements caused by them.
Monday, May 1, 2023
Derived Mechanical SI Units: UNIZOR.COM - Physics4Teens - Units in Physi...
Notes to a video lecture on http://www.unizor.com
For a short introduction to the International System of units (SI) see the previous lecture "SI Intro & Time" within this topic.
Mechanics Units
All units used in the science of Mechanics are derived from base units introduced in the previous chapter Base Units of this part Units in Physics of the course.
Area and Volume
An area of a rectangle is a product of it's two linear dimensions usually called length and width or similarly.
The unit of measure of an area in SI is an area of a square with dimensions 1 meter (m) by 1 meter (m), and it's called a square meter. Its abbreviation is m²
A volume of a right parallelepiped is a product of it's three linear dimensions usually called length, width and height or similarly.
The unit of measure of a volume in SI is a volume of a cube with dimensions 1 meter (m) by 1 meter (m) by 1 meter (m), and it's called a cubic meter. Its abbreviation is m³
Plane and Solid Angles
All central angles with the same ratio of a length of a circle's arc they intercept to a length of a circle's radius are congruent to each other regardless of a radius of a circle. Hence, this ratio can be used as a measure of a plane angle.
A plane angle unit in SI is a central angle in a circle whose corresponding arc's length equals to a length of a circle's radius. This unit is called a radian. Its abbreviation is rad.
Examples:
(1) Full rotation angle corresponds to an arc of the length of 2π·R. Therefore, its measure in radians equals to
2π·R/R = 2π (rad).
(2) Right angle corresponds to an arc of the length of a quarter of the circumference of a circle, that is π·R/2. Therefore, its measure in radians equals to
π·R/(2R) = π/2 (rad).
Radian is considered a derived unit because its related to a linear characteristic - a radius of a circle. However, since we measure plane angles by a ratio between a length of an arc in meters to a length of a radius in meters, that is the unit is m/m, plane angles are called "dimensionless".
Let's recall that a solid central angle is space inside a cone with vertex coinciding with a sphere's center.
All solid central angles with the same ratio of an area of a sphere's cap they intercept to a square of a radius are congruent to each other regardless of a radius of a sphere. Hence, this ratio can be used as a measure of a solid angle.
A solid angle unit in SI is a solid central angle in a sphere whose corresponding cap's area equals to a square of a sphere's radius. This unit is called a steradian. Its abbreviation is sr.
Examples:
(1) Full spherical angle is subtended by a full surface of a sphere of the area of 4π·R². Therefore, its measure in steradians equals to
4π·R²/R² = 4π (sr).
(2) Consider a sphere in the space with Cartesian coordinates with the sphere's center coinciding with a the origin of coordinates.
A solid angle subtended by part of a surface of the sphere with all non-negative coordinates has an area of 1/8th of an area of a full sphere's surface, that is 4π·R²/8=π·R²/2. Therefore, its measure in steradians equals to
π·R²/(2R²) = π/2 (sr).
Steradian is considered a derived unit because its related to a linear characteristic - a radius of a sphere. However, since we measure solid angles by a ratio between an area of a cap in square meters to a square of a radius in square meters, that is the unit is m²/m², solid angles are called "dimensionless".
Speed
Speed, generally speaking, is a distance in units of length (meters in SI) covered by a moving point-object per unit of time (seconds).
This definition is fine for a uniformly moving object, when a distance covered in any period of time is proportional to an amount of time with constant coefficient of proportionality.
If a point-object's motion is not uniform, our definition must be more precise.
We should speak about a speed at any moment of time t.
To measure it, we measure the distance Δx(t) covered by a point-object from this moment t to the next one t+Δt, where Δt is infinitesimal increment of time and take a limit of ratio Δx(t)/Δt, when Δt→0.
In any case, constant or variable speed is a ratio of a distance to time this distance was covered.
Therefore, speed in SI in measured in meters per second (m/s). This unit has no special name and used as a combination of units of distance and time.
Acceleration
Acceleration, generally speaking, is an increment of speed per unit of time.
We rarely study the motion with variable acceleration, but in any case, we have to divide an increment of speed, which is measured in m/s by time in seconds (s).
Therefore, acceleration is measured in meters per second per second, abbreviated as m/s². This unit has no special name and used as a combination of units of distance and time.
Force
The First Newton's Law states that in absence of force an object moves along a straight line with constant speed.
Constant speed means no acceleration.
The Second Newton's Law states that it's the force that causes an object to accelerate, and the acceleration is proportional to a force with a coefficient of proportionality being the mass of an object according to a formula
F = m·a
where
F - force,
m - mass,
a - acceleration
Therefore, we can choose a unit of force as something that is measured using the units of mass and acceleration.
A unit of force that makes a mass of 1 kilogram (kg) to accelerated with 1 meter per second per second (m/s²) is the unit of choice for force in SI and it's called a newton and abbreviated as N.
1N = 1kg·m/²
Pressure
A pressure is an effect of a force onto a surface, like an effect of our body's weight (a force of gravitation) on a floor when we stand still or an effect of particles of air bombarding any surface touched by these particles.
Since a force is measured in newtons (N) and an area is measured in square meters (m²), the measure of pressure is newton per square meter (N/m²). This unit is called pascal and abbreviated as Pa.
1Pa = 1N/m² =
= 1kg·m/(s²·m²) = 1kg/(s²·m)
Density
Density of a substance is its mass per unit of volume.
Since a mass is measured in kilograms (kg) and a volume is measured in cubic meters (m³), the measure of density is kg per cubic meter (kg/m³). This unit has no special name and used as a combination of units of mass and volume.
For a short introduction to the International System of units (SI) see the previous lecture "SI Intro & Time" within this topic.
Mechanics Units
All units used in the science of Mechanics are derived from base units introduced in the previous chapter Base Units of this part Units in Physics of the course.
Area and Volume
An area of a rectangle is a product of it's two linear dimensions usually called length and width or similarly.
The unit of measure of an area in SI is an area of a square with dimensions 1 meter (m) by 1 meter (m), and it's called a square meter. Its abbreviation is m²
A volume of a right parallelepiped is a product of it's three linear dimensions usually called length, width and height or similarly.
The unit of measure of a volume in SI is a volume of a cube with dimensions 1 meter (m) by 1 meter (m) by 1 meter (m), and it's called a cubic meter. Its abbreviation is m³
Plane and Solid Angles
All central angles with the same ratio of a length of a circle's arc they intercept to a length of a circle's radius are congruent to each other regardless of a radius of a circle. Hence, this ratio can be used as a measure of a plane angle.
A plane angle unit in SI is a central angle in a circle whose corresponding arc's length equals to a length of a circle's radius. This unit is called a radian. Its abbreviation is rad.
Examples:
(1) Full rotation angle corresponds to an arc of the length of 2π·R. Therefore, its measure in radians equals to
2π·R/R = 2π (rad).
(2) Right angle corresponds to an arc of the length of a quarter of the circumference of a circle, that is π·R/2. Therefore, its measure in radians equals to
π·R/(2R) = π/2 (rad).
Radian is considered a derived unit because its related to a linear characteristic - a radius of a circle. However, since we measure plane angles by a ratio between a length of an arc in meters to a length of a radius in meters, that is the unit is m/m, plane angles are called "dimensionless".
Let's recall that a solid central angle is space inside a cone with vertex coinciding with a sphere's center.
All solid central angles with the same ratio of an area of a sphere's cap they intercept to a square of a radius are congruent to each other regardless of a radius of a sphere. Hence, this ratio can be used as a measure of a solid angle.
A solid angle unit in SI is a solid central angle in a sphere whose corresponding cap's area equals to a square of a sphere's radius. This unit is called a steradian. Its abbreviation is sr.
Examples:
(1) Full spherical angle is subtended by a full surface of a sphere of the area of 4π·R². Therefore, its measure in steradians equals to
4π·R²/R² = 4π (sr).
(2) Consider a sphere in the space with Cartesian coordinates with the sphere's center coinciding with a the origin of coordinates.
A solid angle subtended by part of a surface of the sphere with all non-negative coordinates has an area of 1/8th of an area of a full sphere's surface, that is 4π·R²/8=π·R²/2. Therefore, its measure in steradians equals to
π·R²/(2R²) = π/2 (sr).
Steradian is considered a derived unit because its related to a linear characteristic - a radius of a sphere. However, since we measure solid angles by a ratio between an area of a cap in square meters to a square of a radius in square meters, that is the unit is m²/m², solid angles are called "dimensionless".
Speed
Speed, generally speaking, is a distance in units of length (meters in SI) covered by a moving point-object per unit of time (seconds).
This definition is fine for a uniformly moving object, when a distance covered in any period of time is proportional to an amount of time with constant coefficient of proportionality.
If a point-object's motion is not uniform, our definition must be more precise.
We should speak about a speed at any moment of time t.
To measure it, we measure the distance Δx(t) covered by a point-object from this moment t to the next one t+Δt, where Δt is infinitesimal increment of time and take a limit of ratio Δx(t)/Δt, when Δt→0.
In any case, constant or variable speed is a ratio of a distance to time this distance was covered.
Therefore, speed in SI in measured in meters per second (m/s). This unit has no special name and used as a combination of units of distance and time.
Acceleration
Acceleration, generally speaking, is an increment of speed per unit of time.
We rarely study the motion with variable acceleration, but in any case, we have to divide an increment of speed, which is measured in m/s by time in seconds (s).
Therefore, acceleration is measured in meters per second per second, abbreviated as m/s². This unit has no special name and used as a combination of units of distance and time.
Force
The First Newton's Law states that in absence of force an object moves along a straight line with constant speed.
Constant speed means no acceleration.
The Second Newton's Law states that it's the force that causes an object to accelerate, and the acceleration is proportional to a force with a coefficient of proportionality being the mass of an object according to a formula
F = m·a
where
F - force,
m - mass,
a - acceleration
Therefore, we can choose a unit of force as something that is measured using the units of mass and acceleration.
A unit of force that makes a mass of 1 kilogram (kg) to accelerated with 1 meter per second per second (m/s²) is the unit of choice for force in SI and it's called a newton and abbreviated as N.
1N = 1kg·m/²
Pressure
A pressure is an effect of a force onto a surface, like an effect of our body's weight (a force of gravitation) on a floor when we stand still or an effect of particles of air bombarding any surface touched by these particles.
Since a force is measured in newtons (N) and an area is measured in square meters (m²), the measure of pressure is newton per square meter (N/m²). This unit is called pascal and abbreviated as Pa.
1Pa = 1N/m² =
= 1kg·m/(s²·m²) = 1kg/(s²·m)
Density
Density of a substance is its mass per unit of volume.
Since a mass is measured in kilograms (kg) and a volume is measured in cubic meters (m³), the measure of density is kg per cubic meter (kg/m³). This unit has no special name and used as a combination of units of mass and volume.
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