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.
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