At time t =0 a grinding wheel has an angular velocity of 30.0 rad/s . It has a c

Question

At time t=0 a grinding wheel has an angular velocity of 30.0 rad/s . It has a constant angular acceleration of 30.0 rad/s^2 until a circuit breaker trips at time t = 1.90 s . From then on, the wheel turns through an angle of 431 rad as it coasts to a stop at constant angular deceleration.

Part A

Through what total angle did the wheel turn between t=0 and the time it stopped?

Express your answer in radians.

______________________________ rad

Part B

At what time does the wheel stop?

Express your answer in seconds.

______________________________ s

Part C

What was the wheel's angular acceleration as it slowed down?

Express your answer in radians per second per second.

____________________________ rad/s^2

Explanation / Answer

Here ,

initial angular speed , wi = 30 rad/s

angular acceleration , a = 30 rad/s^2

t = 1.90 s

part A)

total angle rotated ,

theta = wi * t + 0.5 * a*t^2 + theta0

theta = 30 * 1.9 + 0.5 * 30 * 1.9^2 + 431

theta = 542.15 rad

total angle rotated is 542.15 rad

part B)

after the acceleration , final angular speed ,

w = 30 + 30 * 1.9

w = 87 rad/s

let the time taken to stop is t

and the angular acceleration duing slowdown is a

using third equation of motion

-87^2 + 0^2 = 2 * a * 431

a = - 8.78 rad/s^2

Using seond equation of motion

theta = wi * t + 0.5 *a * t^2

431 = 87 * t - 0.5 * 8.78 * t^2

solving for t

t = 9.82 s

the time taken for the wheel to stop is 9.82 s

c)

the angular acceleration during the deceleration is -8.78 rad/s^2

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