A vinyl record on a turntable rotates at$\text{33}\frac{\text{1}}{\text{3}}\text{rev/min}$. (a) What is its angular speed in radians per second? What is the linear speed of a point on the record (b)$\text{15\hspace{0.17em}cm}$and (c)$\text{7.4\hspace{0.17em}cm}$from the turntable axis?

$\omega =33\frac{1}{3}\text{rev/min}$

For smaller angular displacements, the linear velocity can be described as the product of the angular velocity and the radius of the circular path in which the Here particle is traveling.Use the formula for linear velocity in terms of radius and angular speed. Linear velocity is written as the product of radius and angular velocity.

$\mathbf{v}\mathbf{=}\mathbf{r\omega }$

where, vis velocity, r is radius and$\mathbf{\omega }$ is angular velocity.

Here, first convert $33\frac{1}{3}\text{rev}/\text{min}$into rad/sec as follows:

$\omega =\frac{100}{3}\text{\hspace{0.17em}rev/min}\times \frac{0.1047\text{\hspace{0.17em}rad}/\text{sec}}{1\text{rev/min}}$

$\omega =3.5\text{\hspace{0.17em}rad}/\text{s}$

Hence,angular speed is $3.5\text{\hspace{0.17em}rad/s}$.

Now, use the following formula to find linear speed:

$v=r\omega$

$v=0.15\text{\hspace{0.17em}m}\times 3.5\text{\hspace{0.17em}rad}/\text{s}$

$v=0.52\text{\hspace{0.17em}m}$

Hence,linear speed at $15\text{\hspace{0.17em}cm}$is$52\text{\hspace{0.17em}cm/s}$ .

Now, use the following formula to find linear speed:

$v=0.074\text{\hspace{0.17em}m}\times 3.5\text{rad}/\text{s}$

.$v=0.26\text{\hspace{0.17em}m}/\text{s}$

Hence, linear speed at $7.4\text{\hspace{0.17em}cm}$is .$26\text{m}/\text{s}$