Question

A vinyl record that is initially turning at \(33 \frac{1}{3}\) rpm slows uniformly to a stop in a time of \(15.0 \mathrm{~s}\). How many rotations are made by the record while stopping?

Short answer

Answer: 25 rotations.

Step-by-step solution

Convert rpm to rad/s

First, we need to convert the initial angular velocity from rpm (revolutions per minute) to rad/s (radians per second). To do this, we can use the following conversion factor: \(1 \,\text{rpm} = \frac{2\pi}{60} \, \text{rad/s}\): $$ \omega_0 = 33\frac{1}{3} \, \text{rpm} = 33\frac{1}{3} \cdot \frac{2\pi}{60} \, \text{rad/s} $$ Now, calculate the initial angular velocity: $$ \omega_0 = \frac{100}{3} \cdot \frac{\pi}{30} = \frac{10\pi}{9} \, \mathrm{rad/s} $$

Determine the final angular velocity

Since the vinyl record comes to a complete stop, the final angular velocity, \(\omega\), is \(0 \, \text{rad/s}\).

Calculate the angular acceleration

We know the initial and final angular velocities and the time taken. We can use this information to calculate the uniform angular acceleration, denoted as \(\alpha\). The formula to calculate the angular acceleration is: $$ \alpha = \frac{\omega - \omega_0}{t} $$ Plug in the values: $$ \alpha = \frac{0 - \frac{10\pi}{9}}{15.0\,\mathrm{s}} = -\frac{2\pi}{27} \,\mathrm{rad/s^2} $$

Calculate the angular displacement

Next, we need to find the angular displacement, \(\theta\). We can use the following formula, which relates angular displacement to initial angular velocity, time, and angular acceleration: $$ \theta = \omega_0t + \frac{1}{2}\alpha t^2 $$ Plug in the values: $$ \theta = \left(\frac{10\pi}{9}\right)(15.0\,\mathrm{s}) + \frac{1}{2}\left(-\frac{2\pi}{27}\right)(15.0\,\mathrm{s})^2 = 50\pi\, \mathrm{rad} $$

Convert the angular displacement to the number of rotations

To find the number of rotations, divide the angular displacement by \(2\pi\) radians (as there are \(2\pi\) radians in one full rotation): $$ \text{Rotations} = \frac{\theta}{2\pi} = \frac{50\pi}{2\pi} = 25 $$ So, the vinyl record makes 25 rotations while stopping.