Question

A wheels starts from rest and speeds up to 1500 rpm in 120 s. a) Find the angular acceleration of the wheel assumed constant. b) Find the number revolutions it makes during this time. c) Find the time it takes for the wheel to complete its 10th complete revolution. ( hint: Find angular velocity at the end 9 revolution, which becomes W1 for the 10th revolution) Remember there are 2 radius radians in a revolution.

Answer 1

Part A.

Using 1st rotational kinematic equation:

wf = wi + alpha*t

alpha = (wf - wi)/t

wi = 0 rad/sec

wf = 1500 rpm = 1500*2*pi/60 = 157.08 rad/sec

t = 120 sec

So,

alpha = angular acceleration = (157.08 - 0)/120 = 1.31 rad/sec^2

Part B.

Using 2nd rotational kinematic equation:

theta = w0*t + (1/2)*alpha*t^2

Using above known values:

theta = 0*120 + (1/2)*1.31*120^2 = 9432 rad

theta = 9432/(2*pi) = 1501 rev

Part C.

First find the angular velocity when wheel completes 9 rev

Using 3rd rotational kinematic equation:

wf^2 = wi^2 + 2*alpha*theta

theta = 9 rev = 9*2*pi rad

So,

wf = sqrt (0^2 + 2*1.31*9*2*pi) = 12.17 rad/sec

Now Using 2nd kinematic equation:

theta = wi*t + (1/2)*alpha*t^2

wi = initial angular velocity after 9th revolution = 12.17 rad/sec

theta = when wheel completes 9th to 10 10th rev = 1 rev = 2*pi rad

So,

2*pi = 12.17*t + (1/2)*1.31*t^2

Solving above quadratic equation:

t = 0.503 sec = time taken to complete 10th rev

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