Unizor - Physics4Teens - Energy - Kinetic Energy - Introduction

Notes to a video lecture on http://www.unizor.com



Kinetic Energy - Introduction



Kinetic energy is a quantitative characteristic of an object's motion,
that signifies that this motion can result in some work, when our object
interacts with surrounding environment.



As an example, consider an object of mass m uniformly moving within some inertial frame along a straight line trajectory with speed v.



Assume that at some moment of time there appears a constant force F that acts against its motion. For example, a constant air resistance.



As a result of this interaction with air molecules, our object will slow
down because of air resistance acting against its motion until it
stops. So, the motion of our object caused some work - moving molecules
of air away from the trajectory, after which the motion of our object no
longer exists, it stops completely.

Obviously, instead of air resistance, we can consider friction or
gravity, or any other force, assumed constant for this experiment.



Let's calculate the work done by this force, as it acts against the motion of our object and causes its deceleration.



First of all, we calculate the deceleration a:

a = F/m

Then, knowing deceleration, we get the time t our force acts until an object stops:

t = v/a = m·v/F

Now the distance S our object travels until a full stop:

S = a·t²/2 = F·t²/(2m) =

= F·m²·v²/(2m·F²) =

= m·v²/(2F)

Finally, the work W performed by our force F during the distance S:

W = F·S = m·v²/2



What's most remarkable about this formula is that the work performed by force F does not depend on this force, but only on the characteristic of an object (mass m) and its motion (speed v).



So, it appears that there is something specific for an object (its mass) and its motion (speed) - the amount of work to bring this object to a state of rest, and this quantity of work equals to m·v²/2.

This quantitative characteristic of an object in motion is called its kinetic energy.



Let's slightly complicate the problem. The constant force F slows down our object not to a full stop, but to final speed v_end. What will be the work force F would perform?



As before,

a = F/m

t = (v−v_end)/a = m·(v−v_end)/F

S = v·t − a·t²/2 =

= m·v·(v−v_end)/F −

− F·m²·(v−v_end)²/(2m·F²) =

= m·v·(v−v_end)/F −

− m·(v−v_end)²/(2F) =

= (m/2F)·(v²−v²_end)

Finally, the work W performed by our force F during the distance S:

W = F·S = m·(v²−v²_end)/2 =

= m·v²/2 − m·v²_end)/2



This formula indicates that the work needed to change the state of a moving object from a state with one value of its kinetic energy to another equals to a difference between the values of its kinetic energy at these two states.



Absolutely analogous calculations can prove that the work of a constant force that accelerates an object of mass m from a state of rest to speed v equals to

W = m·v²/2 for any force,

and the work needed to accelerated an object from speed v_beg to v equals to

W = m·v²/2 − m·v²_beg/2



In other words, work, that results from the interaction of moving object with surrounding environment, and its kinetic energy are intimately related. One can be converted into another and vice versa.



Work performed by different forces is, by definition, additive.

It means that, if force F₁ acted on a distance S₁ performing work W₁=F₁·S₁ and force F₂ acted on a distance S₂ performing work W₂=F₂·S₂, then the total amount of work performed by both forces is a sum of their individual work.



Immediate consequence of this is that kinetic energy of a system of objects, each having its own mass and moving with its own speed, equals to a sum of their individual kinetic energies.

That is, kinetic energy is additive.



If N objects of masses m₁, m₂, ... m_N are moving with speeds v₁, v₂, ... v_N then their total kinetic energy equals to

W = Σ_i∈[1,N](m_i·v²_i)/2



As a final example, consider a case when an object of mass m initially moves along the X-axis. The X-component of its velocity vector is V_x and its Y-component of a velocity vector V_y is zero. Then its speed V (a scalar) equals to its X-component V_x (a scalar) and its kinetic energy is

E_k = m·V²/2

where V = V_x



Assume that the force F acts at an angle φ to the X-axis.

In this case we can represent the vector of force F as a sum of two perpendicular vectors of force: F_x acting along the X-axis and F_y acting along the Y-axis.

Obviously,

F = F_x + F_y



These two perpendicular to each other forces, applied to our object,
give it a vector of acceleration that also can be represented as a sum
of two perpendicular vectors

a_x = F_x /m

a_y = F_y /m



Let's assume that the force F acts for a duration of time t.

During this time the force F
performs certain work and the velocity of an object will change. We
will compare the amount of work performed by the force with a change in
the kinetic energy of an object.



Considering the initial speed of an object along the X-axis was V_x and acceleration a_x, the distance covered by our object along the X-axis equals to

S_x = V_x·t + a_x·t² /2

At the same time our object moved along the Y-axis with initial speed 0 and acceleration a_y, covering the distance

S_y = a_y·t² /2



In terms of components of the force F the distances along the coordinates are

S_x = V_x·t + F_x·t² /(2m)

S_y = F_y·t² /(2m)



The total work performed by force F , considering work is additive and can be summarized in each direction, is

W = F_x·S_x + F_y·S_y =

= F_x·V_x·t + F_x²·t²/(2m) +

+ F_y²·t²/(2m)



Now let's calculate the kinetic energy of an object at the end of time period t.

The X-component of the velocity will be equal to

V_x^t = V_x + a_x·t = V_x + F_x·t /m

The Y-component of the velocity will be equal to

V_y^t = a_y·t = F_y·t /m



The object's kinetic energy at the end of the time period t, equals to

E_k^t = m·(V^t)² /2

where V^t is the speed at time t.
Using the Pythagorean Theorem, we can represent (V^t)² as

(V^t)² = (V_x^t)² + (V_y^t)².
Therefore, the kinetic energy can be summarized from X- and Y-components and can be calculated as a sum of:

E_x^t = m·(V_x^t)² /2

E_y^t = m·(V_y^t)² /2



The increment of the kinetic energy is

ΔE = E_x^t + E_y^t − E_k

All we have to do is to compare amount of work W performed by force F and increment of kinetic energy ΔE and make sure that they are equal.



Indeed,

ΔE = m·(V_x+F_x·t /m)²/2 +

+ m·(F_y·t /m)²/2 − m·V_x²/2 =

= F_x·V_x·t + F_x²·t²/(2m) +

+ F_y²·t²/(2m) = W



Therefore, the work performed by a force acting on an object during
certain period of time equals to an increment of a kinetic energy of
this object.

Unizor - Physics4Teens - Mechanics - Problems on Power

Notes to a video lecture on http://www.unizor.com



Problems on Mechanical Power



Problem A



A car engine accelerates a car of mass m from the state at rest to speed V during the time T with constant acceleration along a straight line.

Ignore loss of mass due to burning fuel.

(a) What is the power of the car engine as a function of time?

(b) If engine's power is proportional to amount of fuel supplied to it
in a unit of time (fuel burring speed), how the fuel burning speed will
change over time to assure the kind of motion described in this problem?


Solution:



(a) a = V/T

F = m·a = m·V/T

S(t) = a·t²/2 = (V/T)·t²/2

W(t) = F·S(t) = (m·V/T)·(V/T)·t²/2 = m·V²·t²/(2T²)

P(t) = dW(t)/dt = (m·V²/T²)·t



(b) The fuel burning speed is linearly increasing with time. The faster a
car goes, as it accelerates, - the more power should be supplied to
assure the constant acceleration, and the faster fuel is burning.





Problem B



A car engine supplies constant power P to a car of mass m.

Find its speed and acceleration as a function of time.

Is speed linearly increasing with time?



Solution



We will use the general formula for power as a function of mass, speed and acceleration:

P(t) = m·V(t)·a(t)

or, since a(t)=dV(t)/dt,

P(t) = m·V(t)·dV(t)/dt

Therefore,

P(t)·dt = m·V(t)·dV(t)

In our case P(t) is constant P. Therefore,

P·dt = m·V(t)·dV(t)

Integrating this by time from t=0 to t,

P·t = m·V²(t)/2

From this we can find speed V(t):

V(t) = √2P·t/m

Acceleration is

a(t) = dV(t)/dt = √P·/(2m·t)

Speed is not linearly increasing with time, it is proportional to a
square root of time, which means that acceleration is monotonically
diminishes, while speed increases.





Problem C



A car engine supplies constant power P to a car of mass m.

How long would it take for a car to reach speed V_max?



Solution



As we know,

P(t) = m·V(t)·a(t)

Since power P(t)=P - is constant, and acceleration is the first derivative of speed by time, this can be written as

P = m·V(t)·dV(t)/dt

Therefore,

P·dt = m·V(t)·dV(t)

Integrating this by time from t=0 to t,

P·t = m·V²(t)/2

From this we can find time t as a function of speed:

t = m·V²(t)/(2P)

Therefore, the time at the moment the speed is equal to V_max equals to

t_max = m·V²_max /(2P)

Obviously, the more powerful an engine is - the shorter will be the time interval it takes to achieve needed speed.



As an example, consider a car 2012 Tesla Model S 85 kWh .

It has a mass of 2108 kg and its engine develops the power of 310 Kw.

According to the formula above, the time it takes for this car to reach the speed of 60 miles/hour (26.8 m/sec) equal to

2108·26.8²/(2·310000) = 2.44(sec)

So, in theory, this car is capable to reach the speed of 60 miles/hour in just under 2.5 sec.

Unizor - Physics4Teens - Mechanics - Power

Notes to a video lecture on http://www.unizor.com



Definition of Mechanical Power



To analyze the motion, we often use a concept of speed.

Let's assume that an object moves in some inertial reference frame, and
the distance covered by it from its initial position along its
trajectory is a function of time S(t).

Recall the definition of speed of an object as the rate, at which this object covers the distance along its motion along a trajectory.

In case of a uniform motion we can simply divide the distance S, covered during time t, by the time t to get the speed:

V = S/t

In case of non-uniform motion the speed changes and at any particular moment of time t an instantaneous speed can be calculated using differentials:

V(t) = dS(t)/dt



To analyze the mechanical work performed to achieve certain results, we often use a concept of power.

Let's assume that something or someone performs certain work and, as the time goes by, the work performed is a function of time W(t).

The power is the rate, at which the work is performed.

If during the time t the work performed is W, we define the average power of the whoever or whatever performs the work as

P = W/t

Most likely, at equal in length but different time intervals the amount
of work performed will be different. For example, when a car starts, its
engine should give a car an acceleration, which requires more work per
unit of time than to maintain a constant speed on a smooth straight
road.

In cases like this we can talk about an instantaneous power as a function of time that can be calculated using differentials:

P(t) = dW(t)/dt



Consider an example of an object in uniform motion against the force of friction with a constant speed V.

The force F that moves it forward must be equal in
magnitude and opposite in direction to the force of friction to maintain
the constant speed. Since the friction is constant, the force F must be constant as well.

The distance S covered as a function of time t is

S(t) = V·t

Therefore, the work performed by the force F during the time t is

W(t) = F·S(t) = F·V·t

From the definition of power follows that the power this force F exhorts is

P(t) = dW(t)/dt = F·V

As an example, the car engine exhorts the same power and consumes the
same amount of gas per unit of time, if the car uniformly moves along a
straight road. This power is used to generate a force sufficient to
overcome the friction of wheels and air resistance.



Consider a more general case, when the motion is not uniform.

Assume, an object of mass m moves as a result of action of force F(t), where t is time. The distance it covers is S(t).

Then during an infinitesimal time interval from t to t+dt the work performed by this force will be

dW(t) = F(t)·dS(t)

Considering the Newton's second law,

F(t) = m·a(t),

where a(t) is acceleration.

Increment of distance is

dS(t) = V(t)·dt,

where V(t) is an object's speed.

Also, by definition of acceleration,

a(t) = dV(t)/dt

Therefore,

dW(t) = (m·dV(t)/dt)·V(t)·dt =

= m·V·dV(t)

Power exhausted by this force is, therefore

P(t) = dW(t)/dt =

= m·V·dV(t)/dt =

= m·V(t)·a(t)



From the definition of power as amount of work per unit of time or, more precisely, the first derivative of work by time

P(t) = dW(t)/dt

follows that the unit of measurement of power is joule/sec called watt.

Expanding the definition of joule as newton·meter,

1 watt = 1J/sec = 1N·m/sec

Obvious extensions of unit of power watt are

kilowatt = 1,000 watt and

megawatt = 1,000,000 watt.



There is an old unit of power called horsepower.

Metric horsepower, derived from lifting up against a force of gravity on Earth a weight of mass 75 kg with a constant speed of 1 m/sec, is related to watt unit as

1(metric HP) =

=75(kg)·9.8(m/sec²)·1(m/sec)≅

≅ 735.5(W)

For historical reasons there is also a mechanical horsepower, defined as 33,000 pound-feet per minute, related to watt unit as

1(mechanical HP) ≅ 745.7(W)



So, a car engine of 200 mechanic horsepower has the power of about 149,140 watt.



Watt, as a unit of measurement, was called in honor of James
Watt, an 18th century Scottish scientist who was one of the first to
research a concept of power, developed steam engines and measured the
power of a horse.



Now let's address the concept of power in a case of rotation with constant angular speed. An example is lifting a bucket of water from a well.

Assume, a bucket of water has a mass m and we lift it with constant linear speed V with an angular speed of the well's wheel ω=V/R, where R - radius of a well's wheel.



Since the speed is constant, the force F that acts on a bucket equals to m·g, where g - acceleration of the free fall.

At the same time, if the wheel is turned by some motor and R is the radius of its shaft, the motor manufacturer provides technical characteristic not only of the power, but also of a torque of a motor.

Remember that the torque equals to

τ = F·R



So, on one hand, we have expressed the power of a motor P in terms of unknown force F and linear speed of a bucket:

P = F·V = F·R·ω

On another hand, we expressed the torque of this motor in terms of the same unknown force F and a radius of its shaft:

τ = F·R

Substituting torque τ for F·R in the formula for power, we can find the relationship between the power of a motor and is torque:

P = τ·ω



Notice the similarity between the formula for power in case of uniform motion along a straight line P=F·V and formula for power in case of rotation with constant angular speed P=τ·ω. Instead of force F in case of straight line motion, we use torque τ for rotation and, instead of linear speed V for straight line motion, we use angular speed ω for rotation.



Let's check this with real data about a particular engine.

Below is a graph representing the power and torque of Ford Motor Company 6.7L Power Stroke diesel V-8.



As you see, the power and torque grow relatively monotonically until
some engine limitations start playing significant role. While in the
area of monotonic growth, we can take a particular angular speed, say,
1400 RPM (revolutions per minute) and see that the power of an engine
equals approximately 174 HP (mechanical horsepower) and the torque is
about 650 Lb-Ft (pound-feet).

Let's check if the relationship between power and torque derived above is held in this case.

First of all, we transform all units into standard physical measures defined in SI:

1 RPM (revolutions per minutes) =

= 2π radian per minute =

= 2π/60 radians/sec =

= 0.1048 rad/sec

1 HP = 745.7 W = 745.7 J/sec

1 Lb-Ft = 1.35582 J

Substituting all the above, we see that

angular speed is equal to

1400·0.1048 = 146.6 rad/sec

power is

174·745.7 = 129,751.8 J/sec

torque equals to

650·1.35582 = 881.3 J



Now we can check the relationship between the power, torque and angular speed

P = τ·ω

Indeed,

881.3·146.6 = 129,198.6

As we see, the difference between power and a product of torque by
angular speed is minimal, attributable to imprecise measurement of the
parameters and graph reading.

Unizor - Physics4Teens - Mechanics - Work - Problems

Notes to a video lecture on http://www.unizor.com



Problems on Mechanical Work



Problem A



An object of mass m is pushed up a plane inclined to horizon at angle φ with constant acceleration a to a height h against a friction with coefficient of kinetic friction μ.

(a) What is the amount of work necessary to achieve this goal?



(b) Consider the following real conditions of this experiment:

m = 10kg(kilograms)

g = 9.8m/sec²(meters per sec²)

h = 0.5m(meters)

φ = 30°(degrees)

μ = 0.2

a = 0.1m/sec²(meters per sec²)

The work W=F·S is measured in N·m=kg·m²/sec² units, called joule and abbreviated as J.

Calculate the work W performed in this experiments in joules.



Solution

Let F be the force that pushes the object up the slope, P=m·g is the object's weight, R is the force of resistance from the friction, S is the distance this object moves to reach the height h.

(a) F−P·sin(φ)−R = m·a

R = P·cos(φ)·μ

W = F·S

S = h/sin(φ)

F = P·sin(φ) + R + m·a =

= m·[g·sin(φ) + g·cos(φ)·μ + a]

W = F·h/sin(φ) =

=m·h[g+g·cot(φ)·μ+a/sin(φ)]

(b)
W = 67J(joules)





Problem B



An object of mass m is pushed by a constant force F down a slope of a plane inclined to horizon at angle φ. Initially, it's at rest, the final speed is V. There is a friction with coefficient of kinetic friction μ.

(a) What is the time t from the beginning to the end of the object's motion?

(b) What is the distance S this object moved until reaching the final speed V?

(c) What is the amount of work W performed by this force?

(d) Consider the following real conditions of this experiment:

F = 20N(newtons)

m = 10kg(kilograms)

g = 9.8m/sec²(meters per sec²)

V = 0.5m/sec(meters per sec)

φ = 5°(degrees)

μ = 0.7

Calculate the work W performed in this experiments in joules, distance S and time t of motion.



Solution

P=m·g is the object's weight,

R = P·cos(φ)·μ is the force of resistance from the friction,

S is the distance this object moves to reach the speed V,

a is the acceleration of this object.

Then from the Newton's Second Law

F+P·sin(φ)−R = m·a

(we added P·sin(φ) because an object moves down a slope)

Now we can find an acceleration of our object:

F+m·g·sin(φ)−m·g·cos(φ)·μ =

= m·a

a = F/m + g·sin(φ) − g·cos(φ)·μ

Knowing acceleration a and the correspondence between acceleration, final speed and time V=a·t, we can determine time:

(a) t = V/a

Now we can find the distance

(b) S = a·t²/2 = V·t/2

The work performed by the force F equals to

(c) W = F·S

(d) a = 0.066(m/sec²)

t = 7.576(sec)

S = 1.894(m)

W = 37.879(joules))





Problem C



An object of mass m is dropped down from a certain height above a surface of some planet. At the moment it hits the ground its speed is V.

What is the work W performed by the force of gravity?



Solution

Let the acceleration of free falling on this planet is a, the time of falling t and the height S.

Then

F = m·a

V = a·t

t = V/a

S = a·t²/2 = V²/(2·a)

W = F·S = m·V²/2

Notice, the work depends only on mass and final speed - the results of
action, not on the height, nor acceleration of free falling, nor on
time.

Unizor - Physics4Teens - Mechanics - Work - Golden Rule of Mechanics

Notes to a video lecture on http://www.unizor.com



Golden Rule of Mechanics



Let's summarize what we have learned about mechanical work in the previous lecture.



1. In a simple case of motion along a straight line with a constant force F acting along a trajectory, the most important parameter that quantifies the result of the action is work defined as W=F·S, that fully characterizes and is fully characterized by speed V of an object:

W = F·S = m·V²/2

In particular, it means that increasing the force by a factor of N and decreasing the distance it acts by the same factor of N would result in the same final speed of an object.

So, we can "win" in distance, but we will "lose" in force and vice versa.

This is the first example of the Golden Rule of Mechanics - there
are many ways to achieve the result, you can reduce your distance, but
you will have to increase the force or you can reduce the force, but you
will have to increase the distance.

In short, as we mentioned above, whatever you win in distance you lose in force and vice versa.



2. In case of a constant force acting at an angle to a straight line trajectory, the difference is only a factor cos(φ), where φ is an angle between a force and a direction of a trajectory. In vector form it represents the scalar product (F·S).

So, the Golden Rule works exactly as above.



3. Recall the formula for work of a force F pushing an object of weight P up along an inclined plane of the length S making angle φ with horizon to the height H:

W = F·S = P·H

We've proven this in the previous lecture and, as you see, it's
independent of the slope of an inclined plane. However, the minimum
effort we have to apply as a force to move an object up the slope is F=P·sin(φ), while the distance equals to S=H/sin(φ).

We can reduce the effort (the force F) by using an incline of a smaller slope, but that would lengthen the distance S we have to push an object.

So, again, we see the Golden Rule in action.



4. Consider lifting some heavy object of the weight P using a lever, applying the force F to the opposite to an object end of the lever.



This is a case of equilibrium in rotational motion and the balance can
be achieved by equating the moments of two forces acting against each
other:

F·L_f = P·L_p

If S_f is the distance the force F acts down and S_p is the distance our object moves up,

S_f/S_p = L_f/L_p and

F·S_f = P·S_p

By using a lever with longer arm L_f for the application of force, we can proportionally reduce the force F
achieving the same result - lifting an object to certain height. An
inverse is true as well - we can shorten the arm and proportionally
increase the force.

In any case, the Golden Rule of Mechanics is observed: "winning"
in force - proportionally "losing" in distance or "winning" in distance"
- proportionally "losing" in force.



All the above examples emphasize the importance of the concept of mechanical work as the quantitative measure and characteristic of the purpose and the result of applying a force. The so-called Golden Rule of Mechanics is just a catchy term that underscores the importance of the concept of work.



Acting with the force to achieve certain goal necessitates performing
certain amount of work that depends on the goal, not on how we achieve
it.

Unizor - Physics4Teens - Mechanics - Work - Definition

Notes to a video lecture on http://www.unizor.com



Definition of Mechanical Work



Why do we apply a force to an object?

Usually, to achieve certain result, like to accelerate it to a certain speed or to move it from one point to another.



Straight line motion with a force acting along a trajectory



Consider an acceleration of a car by pressing the gas pedal. As a
result, a car will reach certain speed and then, in absence of the
traffic lights, disregarding friction and air resistance, it goes by
inertia, maintaining this constant speed.



A car with a more powerful engine will need a shorter distance to reach
certain speed. A car with a weaker engine accelerates on a longer
distance to reach the same speed.

It seems, there is some relationship between the force, the distance it
is applied to an object and the final speed reached by this object as a
result of the acting force.



Let's consider the final speed of a constant force F acting on an object of mass m, initially at rest, acting on certain distance S.



The acceleration of this object, according to the Second Newton's Law is

a = F/m

The time t to cover distance S with acceleration a is based on the formula of Kinematics

S = a·t²/2 and is equal to

t = √2S/a  = √2S·m/F 

From this we derive the final speed at the end of acceleration

V = a·t = (F/m)·√2S·m/F  =

= √2F·S/m 



As we see, the final speed of an object of mass m depends on the product of the force and the distance this force is acting.

This product that characterizes the result of applying a force on a certain distance is called the work performed by a given force acting on a given distance:

W = F·S

We can reduce by half the force and double the distance - the resulting speed will be the same.

Using this definition of work, we can represent the final speed as

V = √2W/m 

Resolving for work as a function of final speed, we obtain

W = m·V²/2



So, given a final speed, we can determine how much work it takes to
achieve it and, given amount of work, we can calculate the speed this
work will cause.



This concept of work is closely tied with such less precisely
defined concepts of result, purpose, effect etc. While we not always can
compare the results or effect of two physical experiments, we can
always compare the work done by the forces participating in these experiments.

Thus, the work can be used to measure the results or effect of physical experiment.



Straight line motion with a force acting at an angle to a trajectory



Consider a slightly more complicated example with force acting at an angle to a trajectory.

A toy train of mass m is pulled by a child, who stands on a side of a track, with a constant force F, so the force acts at an angle to a straight line trajectory, making an angle φ with it.



Representing this force F (in blue on a picture above) as a
sum of two forces (in red), one acting along a track, forcing the train
to speed up, and another acting perpendicularly to a track (force N), which is balanced by the reaction of the track R (in brown), we see that the force that pushes a train forward equals to F·cos(φ), while the force acting perpendicularly to a track can be simply ignored as being balanced by an opposite reaction force.



Exactly the same considerations as above leads us to a more universal formula for relationship between the final speed and the work:

V = √2W/m ,

where W = F·S·cos(φ)



Obviously, if the force acts along a straight line of a trajectory, angle φ is zero, its cosine is 1 and the formula corresponds to the one derived earlier.



Resolving the formula above for work, we obtain

W = m·V²/2

which indicates that the work depends on the result of an action
only (final speed), not the way how we achieve this result, using a
stronger force on a shorter distance or a weaker force on a longer
distance.



Motion against gravity along an inclined plane



Let's consider a completely different goal of a physical experiment and show the importance of a concept of work.

This time our purpose is to lift an object of mass m to a certain height H above the ground along a frictionless inclined plane making angle φ with horizon.



The force F needed for this must be equal to a component R of the weight P of an object along the inclined because the other component of its weight N, perpendicular to an inclined, is balanced by a reaction of the plane.

F = P·sin(φ)

The distance S this force is acting on equals to

S = H/sin(φ)

From this we can derive the work performed by force F along the distance S:

W = F·S = P·sin(φ)·H/sin(φ) =

= P·H



This is quite a remarkable result. The work does not depend on the angle of an incline, only on the height we lift the object and its weight.



As in the previous cases, the work seems to be a characteristic
of the result and does not depend on how we have achieved it, using an
inclined with bigger or smaller slope.



Rotational motion with a force acting tangentially to a trajectory



Our final example is about rotation.

Consider a person starts rotating a carousel of a radius r, having a moment of inertia I, from the state of rest to some angular speed ω with constant angular acceleration α, applying constant force F tangentially to the carousel's rim.

From the rotational dynamics we can determine angular acceleration, knowing torque τ of the force and a moment of inertia of a carousel I:

τ = F·r = I·α

α = F·r/I

Linear acceleration of the point of application of the force equals to r·α.



To achieve the final angular speed ω with angular acceleration α we need time

t = ω/α = ω·I/(F·r)

During this time the point of application of force travels around a circle for a distance of

S = r·α·t²/2 =

= r·(F·r/I)·[ω·I/(F·r)]²/2 =

= I·ω²/(2F)



Multiplying by F both sides, we obtain the relationship between work W=F·S and final angular speed achieved as a result of application of the force:

W = F·S = I·ω²/2

This is a rotational equivalent of the analogous formula for straight line movement derived in the beginning of this lecture.



So, given a final angular speed, we can determine how much work it takes
to achieve it and, given amount of work, we can calculate the angular
speed this work will cause.



Definition of work in simple cases presented above



All the above examples provide a proper basis for the following definition of the work:

Work of the force F acting at an angle φ to trajectory on the distance S is W = F·S·cos(φ).



General case of a motion along a curved trajectory



To be more precise and to cover a case of a curved trajectory, we have
to require that this definition should be applied to infinitesimal
amount of work dW performed on infinitesimal interval dS of trajectory:

dW = F·dS·cos(φ)



Using a concept of scalar product of vectors and considering force and interval as vectors, the same definition can be written as

dW = (F ·dS )



The latter represents the most rigorous definition of work.

Unizor - Physics4Teens - Mechanics - Statics - Problems

 



Notes to a video lecture on http://www.unizor.com



Problems on Equilibrium



Problem 1



Two point-objects of mass M and m are hanging at two opposite ends of a weightless rod of length L.

At what point on a rod should we fix a thread, so the system of a rod with two weights is hanging on this thread in equilibrium?

Does this state of equilibrium imply that a rod is horizontal?



Answer

At distance m·L /(M+m) from the end with object of mass M.



Problem 2



An object of weight W is positioned on a flat surface. The coefficient of static friction is μ.

What is the minimum pulling force P to be applied to a
rope, attached to this object, to start pulling it forward, if the angle
between a rope and a horizontal flat surface is φ?



Answer



P = μ·W / [cos(φ)+μ·sin(φ)]



Problem 3



Consider the following illustration to this problem.


A person of weight W stands on a platform of weight P. Pulleys and ropes are arranged as on this illustration.

What is the force F a person should apply to a rope to keep the whole system in equilibrium?



Solution



Let N be a reaction of a platform onto a person. It pushes the person upward.

If F is a force of a person pulling a rope down, and the
system is in equilibrium, that is all components are at rest and all
forces are balanced, the rope pulls a person up with the same force F, according to the Third Newton's Law.

Since a person is in equilibrium, forces directed upward (N+F) should be equal in magnitude to forces directed downwards (weight of the person W):

N+F = W

The platform is also in equilibrium. The forces that push it down are its weight P and reaction N of a person standing on it. The forces pulling the platform up are tensions of two ropes T on the left and F on the right:

P+N = T+F

Finally, a small pulley is in equilibrium. It's pulled down by two tensions F, from the left and from the right. The tension T pulls it up:

T = 2F

We have three equations with three unknowns: T, N and F, which we have to solve for an unknown force F, with which a person pulls a rope to balance the system:

Substituting T from the last equation into the second one, getting a system of two equations:

N+F = W

P+N = 3F

Solving the first equation for N and substituting into the second:

P+W−F = 3F

Resolving for F:

F = (P+W)/4



Answer

F = (P+W)/4



Problem 4



A ladder is standing at some angle to the floor. It's in equilibrium,
that is it's not slipping down to the floor. What holds it is the static friction between it and the floor.

If the ladder stands almost vertically, the static friction is
greater and it easily holds the ladder in the position. As we move the
bottom of a ladder away from the vertical, there will be a point when static friction will no longer could hold the ladder in place and the ladder would slip down to the floor.

What is the smallest angle the ladder would hold if the coefficient of static friction is μ?



Solution



Consider the following illustration to this problem.


While the reaction of the wall N₁ is smaller than the static friction F, the ladder would hold its position.

The smallest angle between the ladder and the floor will be when these two forces are equal.

To be in equilibrium, vector sum of all forces must be equal to
null-vector and a sum of all angular momentums also must be equal to
null-vector.

Along the horizontal X-axis the sum of all forces equals to zero if the horizontal reaction of the wall N₁ equals to the force of static friction F, which, in turn, equals to a product of the vertical reaction of the floor N₂ by a coefficient of static friction μ:

(a) N₁ = μ·N₂

Along the vertical Y-axis the sum of forces equals to zero if the reaction of the floor N₂ equals to weight of a ladder W:

(b) N₂ = W

To analyze angular momentums, let's choose an axis of rotation. It's
easier to choose an axis perpendicular to both X- and Y-axis that goes
through a point where a ladder touches the floor. In this case angular
momentums of forces N₂ and F are zero because their corresponding radiuses are zero.

Let's assume that the length of the ladder is D and the angle between the ladder and the floor is φ. The magnitude of the angular momentum of the force of weight W will then be W·D·cos(φ)/2. The magnitude of the angular momentum of the reaction of the wall N₁ will be N₁·D·sin(φ). The condition of the sum of angular momentums to be equal to zero results in the equation:

N₁·D·sin(φ) = W·D·cos(φ)/2

The length of the ladder can be canceled:

(c) N₁·sin(φ) = W·cos(φ)/2

Equations (a), (b) and (c) form a system of three equations with three unknowns:

(a) N₁ = μ·N₂

(b) N₂ = W

(c) N₁·sin(φ) = W·cos(φ)/2

From this

tan(φ) = 1/(2μ)



Answer



φ = arctan[1/(2μ)]

Interestingly, the answer does not depend on the weight or the length of the ladder.