To describe the motion of rotating or spinning objects, we need a more natural set of variables than the x's, v's, and a's we have been using so far. For instance, every point on a spinning object has a different velocity, but we can define an angular velocity that is the same for all points. v is proportional to r, so all we really need to do it divide out the r.

Angular variables

Angular displacement (angle): θ = s/r

Angular velocity: ω = v/r

Angular acceleration: α = a_t/r

Note that the angular acceleration is connected to the tangential acceleration, not the centripetal acceleration. There is an angular acceleration only when the rotation rate changes.

If the angular velocity is constant, then the speed of a point on the rotating object is:

v = 2πr / T

where T is the period, the time to go around once.

The angular speed is ω = v/r = 2π / T

These angular variables are vectors, just like their straight-line motion cousins. Which way do they point? Take the fingers of your right hand and curl them the way an object is spinning. Stick your thumb out and you get the direction of the angular velocity. The angular acceleration is in the same direction if the object is speeding up its spin, and in the opposite direction if it's slowing down.

We'll often use clockwise and counterclockwise to indicate direction. As with straight-line motion, we can define the positive direction based on what's convenient in a particular case.

We will focus on rotation about a single axis of rotation, which is analogous to one-dimensional straight-line motion. Basically, if you understand 1-D motion you can do rotation - rotational motion is really just straight-line motion rolled up into a circle.

Displacement, velocity, and acceleration all have rotational equivalents. There are also rotational equivalents of mass, force, Newton's Laws, kinetic energy, momentum, etc. Any equation we used for straight-line motion has a rotational form that can be found by substituting the equivalent rotational variables.

For instance, how are angles, angular velocities, and angular accelerations related? The same way the linear variables are:

Angular velocity is the rate of change of angle

Instantaneous angular velocity: ω = dθ/dt

Average angular velocity = ω_avg = Δθ/Δt

Δθ = ∫ ω dt

Angular acceleration is the rate of change of angular velocity

Instantaneous angular acceleration: α = dω/dt

Average angular acceleration = α_avg = Δω/Δt

Δω = ∫ α dt

These equations relate displacement, velocity, acceleration, and time, and apply under the following conditions:

• the acceleration is constant
• the motion is measured from t = 0
• the fact that several variables are vectors is accounted for with appropriate plus and minus signs

You are on a ferris wheel that is rotating at the rate of 1 revolution every 8 seconds. The operator of the ferris wheel decides to bring it to a stop, and puts on the brake. The brake produces a constant acceleration of -0.11 radians/s².

(a) If your seat on the ferris wheel is 4.2 m from the center of the wheel, what is your speed when the wheel is turning at a constant rate, before the brake is applied?

(b) How long does it take before the ferris wheel comes to a stop?

(c) How many revolutions does the wheel make while it is coming to a stop?

(d) How far do you travel while the wheel is slowing down?

θo = 0 θ = ? ωo = 0.785 rad/s ω = 0 α = -0.11 rad/s2

Solution

(a) The wheel is rotating at a rate of 1 revolution every 8 seconds, or 0.125 rev/s. This is the initial angular velocity. It is often most convenient to work with angular velocity in units of radians/s; doing the conversion gives:

ω = 0.125 rev/s * 2π rad/rev = 0.785 rad/s

Your speed is simply this angular velocity multiplied by your distance from the center of the wheel:

v = r ω = 4.2 * 0.785 = 3.30 m/s

(b) We've calculated the initial angular velocity, the final angular velocity is zero, and the angular acceleration is -0.11 rad/s². This allows the stopping time to be found:

ω = ω_o + α t

t = (ω - ω_o) / α

t = (0 - 0.785)/(-0.11) = 7.14 s

(c) One way to find the number of revolutions the wheel undergoes as it slows to a stop is to find the angle it moves through:

θ - θ_o = ω_o t + ½ α t²

θ = (0.785 * 7.14) + ½ (-0.11) * (7.14)² = 2.80 radians

This can be converted to revolutions:

2.80 rad / (2π rad/rev) = 0.446 revolutions.

(d) To figure out the distance you traveled while the wheel was slowing down, the angular displacement (in radians) can be converted to a displacement by multiplying by r:

s = rθ = 4.2 * 2.80 = 11.8 m