A CD with a diameter of 12.0 cm starts from rest and with a constant angular acceleration of 1.0 rad/sec² acquires an angular velocity of 5.0 rad/sec. The CD continues rotating at 5.0 rad/sec for 15.0 seconds and then slows to a stop in 12.0 second with a constant angular acceleration. What is the linear speed of a point 4.0 cm from the center at the time 2.0 seconds from the start? A) 0.060 m/s B) 0.080 m/s C) 0.10 m/s D) 0.14 m/s E) 0.18 m/s

Question

Quizplus · Accepted Answer

Initial angular velocity, $\omega_0 = 0$ (since it starts from rest)

Final angular velocity, $\omega = 5.0 	ext{ rad/s}$

Time taken to reach final velocity, $t_1 = \dfrac{\omega-\omega_0}{\alpha} = \dfrac{5.0}{1.0} = 5.0	ext{ s}$

During this time the angular displacement, $	heta_1 = \dfrac{1}{2}\alpha t_1^2 = \dfrac{1}{2}\cdot1.0\cdot(5.0)^2 = 12.5	ext{ rad}$

Now, the CD continue to rotate at same angular velocity for $t_2 = 15s$. So in this time the angular displacement is given by, $	heta_2 = \omega t_2  = 5.0\cdot 15 = 75	ext{ rad}$

Total angular displacement, $	heta = 	heta_1 + 	heta_2 = 12.5 + 75 = 87.5	ext{ rad}$

At this point CD starts decelerating and finally stops after $t_3 = 12.0	ext{ s}$

Using $v^2 = u^2 +2as$ for angular acceleration we have,

$0^2 = 5.0^2 - 2\alpha s$

$\alpha = \dfrac{5^2}{2	imes0.12} = 104.2 	ext{ rad/s}^2$

Time taken to stop, $t_4 = \dfrac{5.0}{104.2} = 0.048	ext{ s}$

During this time angular displacement, $	heta_4 = \dfrac{1}{2}\alpha t_4^2 = \dfrac{1}{2}\cdot104.2\cdot(0.048)^2 = 0.12	ext{ rad}$

Now we can calculate the angular displacement from t=0s upto t=2s,

$	heta_{total} = 	heta_1 + \omega t_2 + 	heta_4 = 12.5 + 75 + 0.12 = 87.62	ext{ rad}$

Angular velocity at this time, $t=2	ext{ s}$,

$\omega = \omega_0 + \alpha t = 0 + 1.0\cdot2.0 = 2.0	ext{ rad/s}$

Using $v = r\omega$ linear velocity at a point, $r=0.04	ext{ m}$ from the center is,

$v = 0.04	imes 2.0 = 0.08	ext{ m/s}$

Therefore, the linear speed of a point 4.0 cm from the center at the time 2.0 seconds from the start is 0.080 m/s.

A CD with a Diameter of 12