PhysicsLAB: Rotational Kinematics

Consider a disk rotating on a stationary rod.

If we now view this disk from the top,

we see that when the disk rotates so that the arc length (s) equals the length of the radius of the disk (r), the subtended central angle (θ) will equal 1 radian. The resulting equation is

s = rθ

θ = s/r

where the unit of a radian represents the dimensionless measure of the ratio of the circle's arc length to its radius.

If the disk rotates through one complete revolution, then s equals the entire circumference and θ equals 2π radians.

s = rθ
2πr = rθ
θ = 2π radians

Since one complete revolution equals 360º, we now have the conversion that

360º = 2π radians
1 radian = 180/π or approximately 57.3º.

Differentiating the basic equation s = rθ results in the next two relationships between the tangential motion of a point along the circumference and the angular behavior of the disk itself.

s = rθ
ds/dt = r(dθ/dt)
v = rω

The Greek letter ω represents the angular velocity of the disk. It is measured in the unit radians/sec. Differentiating one more time, we have

v = rω
dv/dt = r(dω/dt)
a = rα

The Greek letter α represents the angular acceleration of the disk. It is measured in the unit radians/sec².

Summary

These three equations

s = rθ
v = rω
a = rα

allow us to relate the linear motion (s, v, a) of a point moving in circular motion on a rotating platform with the rotational motion (θ, ω, α) of the platform itself.

It is important when using these equations that the units on s, v, a, and r be consistent. That is, if the radius is measured in meters, then s must also be in meters, v in m/sec, and a in m/sec².

Using our new variables for angular motion, we can now state analogous equations to those we have already used for linear motion. When the velocity is constant our equations are

s = vt which becomes θ = ωt.

The following chart shows the relationships between the equations for uniform linear acceleration and those that deal with uniform angular acceleration. When working with pure rotational motion, the standard unit of measurement is the radian.

linear		angular
a = (v_f - v_o)/t		α = (ω_f - ω_o)/t
v_f = v_o + at		ω_f = ω_o + αt
s = ½(v_f + v_o)t		θ = ½(ω_f + ω_o)t
s = v_ot + ½at²		θ = ω_ot + ½αt²
v_f² = v_o² + 2as		ω_f² = ω_o² + 2αθ

Angular acceleration is considered to be constant in these types of situations:

Any horizontal circular motion with a constant applied torque.
Any vertical circular motion driven by a motor with constant power.
Satellites in circular orbits.
A yo-yo falling straight down.

Angular acceleration not considered to be constant, or uniform, in these situations:

When a vertically rotating rod is swinging on a fixed pivot.
In a vertical circle (pendulum), the instantaneous acceleration depends on θ.
When there is a drag force that depends on the object’s velocity.

When an object's angular acceleration is not constant, it instantaneous final angular velocity be determined by using conservation of energy techniques. Its average velocity can be found using

θ_net = ω_avt

Now let's work some examples.

Refer to the following information for the next three questions.

An old phonograph record revolves at 45 rpm.

	What is its angular velocity in rad/sec?

	Once the motor is turned off, it takes 0.75 seconds to come to a stop. What is its average angular acceleration?

	How many revolutions did it make while coming to a stop?

Refer to the following information for the next two questions.

A fan that is turning at 10 rev/min speeds up to 25 rev/min in 10 seconds.

	How many revolutions does the blade require to alter its speed?

	If the tip of one blade is 30 cm from the center, what is the final tangential velocity of the tip?

Refer to the following information for the next two questions.

Consider a standard analog wall clock with a second hand, minute hand, and hour hand.

	Calculate the angular velocity of the second hand of a clock.

	If the second hand is 8" long (there are 2.54 cm in every inch), what is the linear velocity of the tip of the second hand?

Refer to the following information for the next question.

Two wheels are connected by a common cord. One wheel has a radius of 30 cm, the other has a radius of 10 cm.

	When the small wheel is revolving at 10 rev/min, how fast is the larger wheel rotating?

Refer to the following information for the next question.

A rotor turning at 1200 rev/min has a diameter of 5 cm. As it turns, a string is to be wound onto its rim.

	How long a piece string will be wrapped in 10 seconds?