ROTATIONAL KINEMATICS
Axis of rotation is the center of motion for all parts of a rotating body. It is a line segment within the body about which all parts rotate.
Angular displacement is an expression of the angle through which an object has rotated or revolved. Rotate means to spin about an internal axis, revolve means to move around some external point or axis. The skater's body is rotating. The points A, B, and C are revolving.
Angular displacement can be expressed in revolutions, degrees or radians. The SI unit is the radian.
In the diagram, Ө, Өo, and ΔӨ are examples of angular displacement.
In scientific applications, the radian is often used because this makes the relationship between arc length, radius of the circle, and angular displacement easier to work with.
The equation for this is:
S = RӨ
Where S is arc length, R is the radius of the circle and Ө is the angular displacement when expressed in radians.
If S is the entire circumference of a circle, Ө must be 2Π since
C = 2ΠR
This means that 2Π radians is the same angular measure as 360° or one revolution. We may then convert between units of angular displacement using these relationships.(1 rad = 57.3°)
The diagram above illustrates the relationship between arc length and radius. An observer on earth sees the moon and the sun subtending nearly the same angle even though the sun has a much larger diameter than the moon. Since the sun is about 375 times farther from the earth than the moon, its arc length must be about 375 times larger to subtend the same angle. Since these conditions are met, the moon can almost completely cover the sun by occupying the same area of the sky producing a solar eclipse.
Angular velocity is defined as the time rate of change of the angular displacement. The equation is:
ω = ΔӨ/Δt
where ω is the average angular velocity, ΔӨ is the angular displacement, and Δt is the time interval during which the displacement occurs. Notice that this equation is very similar to the equation for linear velocity.(V = ΔS/Δt)
The SI unit is rad/s.
Angular velocity is a vector and has a direction as any other vector does. When working in two dimensions, the direction may be clockwise or counterclockwise. If the fingers of the right hand are curled in the direction of rotation, the right thumb points in the direction of the angular velocity vector.
Example
A diver completes 3.5 somersaults in 1.7 s. Find the average angular speed in rad/s of the diver.
Example
A device used to measure the speed of a bullet consists of two rotating disks separated by a distance d = 0.85m and having a known angular velocity of 95.0 rad/s. A bullet is fired through the disks and the angular displacement between the two holes is 0.240 rad. Find the speed of the bullet.
Angular acceleration is defined as the time rate of change of the angular velocity. Average angular acceleration is calculated using the equation:
α = Δω/Δt
where α is the angular acceleration, Δω is the change in angular velocity, and Δt is the change in time. Notice that this equation is very similar to the equation for linear acceleration.(a = ΔV/Δt)
The SI unit for angular acceleration is the rad/s2.
We will be working with angular acceleration that is constant or average.
Example
A CD has a playing time of 74 min. When the music starts, the CD rotates at 480 rpm and at the end it is rotating at 210 rpm. Find the average angular acceleration in rad/s2.
The equations of rotational kinematics are very similar to the equations for linear kinematics. Since we have seen that linear displacement along an arc equals the product of angular displacement and the radius of the circle(S = RӨ), linear velocity along the arc equals the product of angular velocity and the radius of the circle(V = Rω), and linear acceleration equals the product of the angular acceleration and the radius of the circle(a = Rα), it follows that the equations of linear kinematics may be used for circular motion. If we convert the linear variables into rotary variables using the three relationships given above, we can use the same equations.
V = Vo + at
ω = ωo + αt
S = ½( Vo + V)t
Ө = ½(ωo + ω)t
S = Vot + ½at2
Ө = ωot + ½αt2
V2 = Vo2 + 2aS
ω2 = ωo2 + 2αӨ
These equations are used in the same way that their linear counterparts are used. Remember these are valid only if the angular acceleration is constant.
The units used may be radians, degrees or revolutions if they are consistent throughout the equation. Unless the answer is specified to be in degrees or revolutions, the radian is the unit of choice.
Example
A toy top at rest has a 64 cm string wrapped around it at a place where the radius of the top is 2.0 cm. The string's thickness is negligible. Find the final angular velocity of the top when the string is pulled completely off with an angular acceleration of +12 rad/s2.
Tangential velocity and acceleration are linear vectors directed along the path of the arc followed by the object as it revolves. Their directions are constantly changing. Their magnitudes can be found using the relationship between each of them and their rotary counterparts.
As you can see from the diagram, the larger R is, the larger the tangential velocity of a skater must be if the angular velocity is the same for all of them. This is also true for the relationship between linear and angular acceleration.
Example
The earth has a radius of 6.38 x 106 m and turns once on its axis every 23.9 hours. (a)Find the tangential speed of a person on the equator in m/s. (b) Find the angle θ in the diagram where the tangential speed is 1/3 of that at the equator.
While studying uniform circular motion we condidered the linear speed of the whirling object to be constant. During the time it is reaching that speed or slowing down, the speed is not constant and a tangential acceleration exists. This type of motion is called nonuniform circular motion and the net acceleration and net force must include both tangential and centripetal components.
Since they are at right angles, the Pythagorean Theorem can be used to determine the net acceleration and then the net force.
Example
A rectangular plate is rotating about an axis that passes perpendicularly through one corner as in the diagram. The tangential acceleration at corner A has twice the magnitude of that at corner B. Find the ratio of the lengths of the two sides of the rectangle.
When an object is rolling we can consider the linear velocity of its center of rotation to have the same magnitude as the tangential velocity of a point on its rim. This is true if there is no slipping.
We can also conclude that the linear distance traveled by its center of rotation is the same as the linear distance traveled by a point on the rim. The equations V = Rω and S = RӨ allow us to relate the linear speed of the rolling object to its angular speed.
Example
A car is traveling at a speed of 20.0 m/s on a straight, level road. The radius of each tire is 0.300 m. If the car speeds up with a linear acceleration of 1.5 m/s2 for 8.0 s, find the angular displacement of each tire(no slipping).
Since rotary motion does not occur in one direction as linear motion can, the direction of rotary motion is expressed differently. If we consider the rotation to be in a plane, the direction of the rotary motion vector can be taken to be at right angles to the plane of rotation.
The two possibilities are clockwise and counterclockwise. Counterclockwise is generally considered to be positiveand clockwise is considered to be negative.
Use your right hand to determine the direction of the vector. If you curl the fingers of your right hand in the direction of rotation and extend your thumb, your thumb will point in the direction of the rotary motion vector.
These vectors may be added and subtracted in the same manner as the linear vectors already discussed.