A student holds a bike wheel and starts it spinning with an initial angular spee

Question

A student holds a bike wheel and starts it spinning with an initial angular speed of 9.0 rotations per second. The wheel is subject to some friction, so it gradually slows down. In the 10-s period following the initial spin, the bike wheel undergoes 65.0 complete rotations. Assuming the frictional torque remains constant, how much more time Delta t_s will it take the bike wheel to come to a complete stop? The bike wheel has a mass of 0.625 kg and a radius of 0.385 m. If all the mass of the wheel is assumed to be located on the rim, find the magnitude of the frictional torque tau_f that was acting on the spinning wheel.

Explanation / Answer

here,

initial angular speed , w0 = 9 rot/s

w0 = 56.52 rad/s

time taken , t1 = 10 s

theta = 65 * 2pi rad = 408.2 rad

let the angular accelration be alpha

theta = w0 * t1 + 0.5 * alpha * t1^2

408.2 = 56.52 * 10 + 0.5 * alpha * 10^2

solving for alpha

alpha = - 3.14 rad/s^2

let the time taken to stop the wheel be t2

w = w0 + alpha * t2

0 = 56.52 - 3.14 * t2

t2 = 18 s

it will take 18 s to stop the wheel

so more time taken , t = t2 - t1 = 8 s

mass , m = 0.625 kg

radius , r = 0.385 m

the magnitude of frictional torque , T = I * alpha

T = m * r^2 * alpha

T = 0.625 * 0.385^2 * 3.14

T = 0.29 N.m

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