Question

An old-fashioned single-play vinyl record rotates on a turntable at

45.0 rpm. What are (a) the angular velocity in rad/s and (b) the period of the motion?

Answer 1

Concept and reason

The concepts used are angular velocity, conversion factor, and the time period of the motion.

The units of the angular velocity are converted by using the conversion factor.

The period of the motion is calculated by using the relation between the angular velocity and the time period.

Fundamentals

Angular velocity of the rotating object is the rate of change of the angular position of the rotating object.

A conversion factor is used to change the units of a measures quantity without changing its value.

The conversion factor for the angular velocity to convert the units from rpm to the rad/s is $\frac{{\left( {\frac{{2\pi }}{{60}}{\rm{ rad/s}}} \right)}}{{1{\rm{ rpm}}}}$ .

The time period is the time taken by the rotating object to complete one complete rotation.

The relation between the time period and the angular velocity is given by,

$T = \frac{{2\pi }}{\omega }$

Here, $\omega$ is the angular velocity and T is the time period.

(a)

Convert the units of angular velocity from rpm to rad/s by using the conversion factor as follows:

$\begin{array}{c}\\\omega = \left( {45.0{\rm{ rpm}}} \right)\left( {\frac{{\left( {\frac{{2\pi }}{{60}}{\rm{ rad/s}}} \right)}}{{1{\rm{ rpm}}}}} \right)\\\\ = 4.71{\rm{ rad/s}}\\\end{array}$

(b)

The expression for the time period T in terms of angular velocity $\omega$ is given by,

$T = \frac{{2\pi }}{\omega }$

Substitute $4.71{\rm{ rad/s}}$ for $\omega$ in above equation as follows:

$\begin{array}{c}\\T = \frac{{2\pi }}{{4.71{\rm{ rad/s}}}}\\\\ = 1.33{\rm{ s}}\\\end{array}$

Ans: Part a

The angular velocity of old-fashioned single-play vinyl record rotates on a turntable is $4.71{\rm{ rad/s}}$ .

Part b

The period of the motion is $1.33{\rm{ s}}$ .