A bug is riding on a music CD at a point 5 cm from the center. The CD starts rot

Question

A bug is riding on a music CD at a point 5 cm from the center. The CD starts rotating at t - 0, and accelerates uniformly, reaching its final rotational speed of 500 rpm in 3 seconds. At t = 1 s, what is the bug's angular acceleration, in (rad)/s2? At t = 1 s, what is the bug's tangential acceleration, in m/s2? At t = 1 s, what is the bug's rotational speed, in (rad)/s? At t = 1 s, what is the bug's linear (tangential) speed, in m/s? At t = 1 s, what is the bug's centripetal acceleration, in m/s2? At t = 4 s, what is the bug's linear (tangential) speed, in m/s? At t = 4 s, what is the bug's tangential acceleration, in m/s2? At t=4s, what is the bug's centripetal acceleration, in m/s2?

Explanation / Answer

Angular accln (alpha) is constant,

Let w be the angular velocity.

1) w_f = w_i + alpha*t

w_i = 0

w_f = 500*2*3.14 /60 = 52.33 rad/s

52.33 = 0 + alpha *3

alpha = 17.44 rad/s^2

Bug's angular velocity after 3 sec = 500*2*3.14 /60 = 52.33 rad/s

Bug's angular accln = [(500*2*3.14/60) - 0 ]/ 3 = 17.44 rad/s^2

2)Tangential accln = angular accln * Radius

Tangential accln, a = alpha*R = 17.44*0.05 = 0.872 m/s^2

3) w = w_i + alpha*t
Rotational speed = 0 + alpha*t = 17.44*1 = 17.44 rad/s

4) v = w * Radius
Tangential speed, v = Rotational speed*R = 17.44*0.05 = 0.872 m/s

5) Centripetal accln = v^2/r = 0.872^2 / 0.05 = 15.20 m/s^2

6) v = w * Radius
Linear speed, v = Rotational speed ( at t = 4s) * R = 52.33*0.05 = 2.6165 m/s

7)Since, bug has zero angular accln after t= 3s, so tangential accln is zero.
Angular accln and tangential accln are same
Tangential accln = 0 m/s^2

8) Centripetal accln = v^2/r = 2.6165^2 / 0.05 = 136.92 m/s^2

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