Please show neatly, hand-written if possible. A turntable is moving with an angu

Question

Please show neatly, hand-written if possible.

A turntable is moving with an angular velocity of coo = 0.4 rad/s. Starting at time t = 0 the turntable is subjected to an angular acceleration of a = 0.2 rad/s^2. What is the angular speed of the turntable at t = 5 s? Through what angle has the turntable rotated at t = 5 s? How long will it take the turntable to make two complete revolutions? The turntable of a record player rotates at the rate of 33 1/3 rev/min and takes 60 s to come to rest when switched off. What is the magnitude of the angular acceleration? How many revolutions does the turntable make before coming to rest?

Explanation / Answer

here,

Ques 1:
initial angular speed, w1 = 0.4 rad/s
angular acceleration, alpha = 0.2 rad/s^2

Part a:
From First eqn of rotationla motion,
wf = wi + alpha *t
wf = 0.4 + 0.2*5
wf = 1.4 rad/s

Part b:
From second eqn of rotational motion,
theta = wi*t + 0.5*alpha*t^2
Theta = 0.4*5 + 0.5*0.2*5^2
theta = 4.5 rad

Part c:
theta = 2 rev = 2 * 2*pi = 12.566 rad

From Seocnd eqn of rotational motion,
theta = wi*t + 0.5*alpha*t^2
12.566 = 0.4*t + 0.5*0.2*t^2
t = 9.4 s

Ques 2:
angular speed, w = 33 rev/min = 33*2*pi/60 = 3.456 rad/s
time, t = 60 s

From first eqn of rotaionla motion,
w/t = alpha
alpha = 3.456/60

The angular acceleration of turntable, alpha = 0.058 rad/s^2

From third eqn of motion, theta = w^2/(2*alpha)
theta = 3.456^2/(2*0.058)
theta = 102.965 rad
or
Theta = 102.965 /(2*pi) = 16.4 rev

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