Constant acceleration equations

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

Straight-line motion	Rotational motion
v = v_o + at	ω = ω_o + α t
x - x_o = v_o t + ½ a t²	θ - θ_o = ω_o t + ½ α t²
x - x_o = ½ (v + v_o) t	θ - θ_o = ½ (ω + ω_o) t
v² = v_o² + 2 a (x - x_o)	ω ² = ω_o² + 2 α (θ - θ_o)

Sample problem

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?

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

θ_o = 0
θ = ?
ω_o = 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