Question

At time t=0 a grinding wheel has an angular velocity of 28.0 rad/s . It has a constant angular acceleration of 28.0 rad/s2 until a circuit breaker trips at time t = 2.00 s . From then on, the wheel turns through an angle of 440 rad as it coasts to a stop at constant angular deceleration. A-Through what total angle did the wheel turn between t=0 and the time it stopped? Express your answer in radians. B-At what time does the wheel stop? C-What was the wheel's angular acceleration as it slowed down?

Answer 1

A)
Initial angular velocity, i = 28 rad/s
Angular acceleration, = 28 rad/s²
Time taken, t = 2 s
Distance traveled, = i t + 1/2 t²
= 28 x 2 + 0.5 x 28 x 4
= 112 rad

The wheel turned another 440 rad before coming to stop.
Total angle = 112 + 440
= 552 rad.

B)
From first part of the motion, i.e, the wheel is accelerating,
Final angular velocity, f = i + t
= 28 + 28 x 2
= 84 rad/s

Now consider the second part of the motion, i.e, the wheel decelerating.
Final angular velocity, 'f = 0
Initial angular velocity, f = 84 rad/s
Distance traveled, = 440 rad
Using the relation, ('f)2 - (f)² = 2
0 - 84² = 2 x 440
= - 8.02 rad/s².

Using the relation
'f = f + t,
0 = 84 + (-8.02) x t
t = 10.48 s.