Question

A 17 g audio compact disk has a diameter of 12 cm. The disk spins under a laser that reads encoded data. The first track to be read is 2.3 cm from the axis; as the disk plays, the laser scans tracks farther and farther from the center. The part of the disk directly under the read head moves at a constant 1.2 m/s. When a disk is inserted, it takes 2.4 s to spin up from rest.
1. What is the torque of the motor?

Answer 1

Answer:

0.00066518 Nm

Explanation:

v = Velocity = 1.2 m/s

r = Distance to head = 2.3 cm

$\omega_f$ = Final angular velocity

$\omega_i$ = Initial angular velocity = 0

$\alpha$ = Angular acceleration

t = Time taken = 2.4 s

Angular speed is given by

$\omega=\dfrac{v}{r}\\\Rightarrow \omega=\dfrac{1.2}{0.023}\\\Rightarrow \omega=52.17391\ rad/s$

From equation of rotational motion

$\omega_f=\omega_i+\alpha t\\\Rightarrow \alpha=\frac{\omega_f-\omega_i}{t}\\\Rightarrow \alpha=\frac{52.17391-0}{2.4}\\\Rightarrow \alpha=21.73912\ rad/s^2$

Torque

$\tau=I\alpha\\\Rightarrow \tau=\dfrac{1}{2}mR^2\alpha\\\Rightarrow \tau=\dfrac{1}{2}0.017\times 0.06^2\times 21.73912\\\Rightarrow \tau=0.00066518\ Nm$

The torque of the motor is 0.00066518 Nm