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

Question

At time t=0 a grinding wheel has an angular velocity of 30.0 rad/s . It has a constant angular acceleration of 34.0 rad/s2 until a circuit breaker trips at time t = 2.50 s . From then on, the wheel turns through an angle of 437 rad as it coasts to a stop at constant angular deceleration.

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

Express your answer in radians.

2) At what time does the wheel stop?

Express your answer in seconds.

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

Express your answer in radians per second per second.

Explanation / Answer

here,

initial velocity of the wheel , w0 = 30 rad/s

angular accelration , a = 34 rad/s ^2

a)

at t = 2.5 s

theta = w0*t + 0.5 * a*t^2

theta = 30 * 2.5 + 0.5 * 34 * 2.5^2

theta = 181.25 rad

the total angle = 181.25 rad + 437 rad = 618.25 rad

the total angle turned by the wheel is 618.25 rad

b)

let the time at which the wheel stops be t'

velocity at t = 2.5 s be w

w = 30 + 34*2.5

w = 115 rad/s

let the angular deaccelration be a'

0 - w^2 = 2*theta'*a'

0 - 115^2 = 2*437 * a'

a' = 15.13 rad/s^2

and

0 = 115 - 15.13 *t'

t' = 7.6 s

the total time taken is 7.6 + 2.5

the wheel will stop at t= 10.1 s

c)

the wheel angular accelration when it slows down is - 15.13 rad/s^2

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