Question

A vinyl record that is initially turning at \(33 \frac{1}{3}\) rpm slows uniformly to a stop in a time of \(15 \mathrm{~s}\). How many rotations are made by the record while stopping?

Short answer

Based on the step by step solution, the short answer is: The vinyl record makes 26.25 rotations while slowing down and stopping.

Step-by-step solution

Identify the given information and symbols to be used

We are given the initial rotational speed, denoted by \(\omega_0\), as \(33\frac{1}{3}\) rpm (rotations per minute). We also know that the final rotational speed, \(\omega_f\), is 0 rpm, since the vinyl comes to a stop. The time taken for the vinyl to slow down and stop is \(t = 15\) seconds. We need to find the number of rotations made by the record while stopping.

Convert rotational speed to radians per second

To find the angular displacement, we need to work in radians and seconds. So, we need to convert the initial rotational speed to radians per second. Use the following conversion factors: \(1\ \text{rotation} = 2\pi\) radians and \(1\ \text{minute} = 60\ \text{seconds}\) \(\omega_0 = 33\frac{1}{3}\ \text{rpm} \cdot \frac{2\pi\ \text{radians}}{1\ \text{rotation}} \cdot \frac{1\ \text{minute}}{60\ \text{seconds}} = 3.5\pi\ \frac{\text{radians}}{\text{s}}\)

Find the angular deceleration

As the vinyl slows to a stop, it experiences an angular deceleration, denoted by \(\alpha\). Since we know the initial and final angular speeds and the time it takes for the change, we can use the equation of motion: \(\omega_f = \omega_0 + \alpha t\) Solve for \(\alpha\): \(\alpha = \frac{\omega_f - \omega_0}{t} = \frac{0 - 3.5\pi}{15} = -\frac{7\pi}{30}\ \frac{\text{radians}}{\text{s}^2}\)

Find the angular displacement

Now that we have the angular deceleration, we can find the total angular displacement using another equation of motion: \(\Delta \theta = \omega_0t + \frac{1}{2}\alpha t^2\) Plug in the values: \(\Delta \theta = 3.5\pi(15) - \frac{1}{2}\left(-\frac{7\pi}{30}\right)(15)^2 = 52.5\pi\) radians

Convert angular displacement to rotations

Now, we just need to convert the angular displacement back to rotations: \(\text{rotations} = \frac{\Delta \theta}{2\pi} = \frac{52.5\pi}{2\pi} = 26.25\) rotations The vinyl record makes 26.25 rotations while stopping.

Key concepts

Rotational Kinematics

When we study the motion of objects that rotate, such as a vinyl record on a turntable, we delve into the realm of rotational kinematics. This field of physics allows us to analyze rotational motion using quantities like angular velocity, angular acceleration, and angular displacement.

What Is Angular Displacement?
Angular displacement, typically denoted by \( \Delta \theta \), is a measure of the angle through which a point or line has been rotated in a specified sense about a specified axis. It is crucial to understand that this is a vector quantity, which means it has a magnitude and a direction.

Converting Units
In the given exercise, the initial rotational speed is provided in revolutions per minute (rpm). For our calculations, we need to convert this into radians per second to align with standard SI units. This is important because radians provide a coherent relation with other rotational quantities and enable the use of trigonometry to analyze motion.

By using the conversion factors \(1\ rotation = 2\pi\ radians\) and \(1\ minute = 60\ seconds\), we make sure our analysis is accurate and standardized, facilitating the application of rotational kinematics equations.

Angular Deceleration

Just as objects can speed up or slow down in linear motion, they can also experience changes in rotational speed. This change is known as angular acceleration if the speed increases, or angular deceleration if it decreases.

Understanding Angular Deceleration
Angular deceleration is the rate at which an object's angular velocity decreases over time. It's vector in nature and has both magnitude and direction; it is directed opposite to the direction of the angular velocity. Whenever you see a rotating object come to a stop, like our vinyl record, it has undergone angular deceleration.

Calculating Angular Deceleration
In your exercise, we derived the angular deceleration (\( \alpha \) by rearranging the motion equation \(\omega_f = \omega_0 + \alpha t\) and solving for \( \alpha \). Being given the initial angular velocity (\( \omega_0 \) and the final angular velocity (\(\omega_f = 0\) since the vinyl stops, along with the time duration, we found that the vinyl experiences a negative acceleration, indicating a slowdown.

This deceleration is crucial because it directly affects the total angular displacement, ultimately telling us how much the vinyl has rotated before it stopped.

Equations of Motion

Equations of motion are the tools physicists use to connect kinematic quantities like displacement, velocity, acceleration, and time. These equations are typically used for linear motion, but they have analogs in rotational motion with corresponding rotational quantities.

Rotational Equations of Motion
Similar to the linear case, where we have equations to relate distance, speed, acceleration, and time, in rotational motion we have equations that relate angular displacement, angular velocity, angular acceleration, and time. These equations are derived from calculus and assume constant angular acceleration.

Application in the Exercise
In the exercise, after finding the angular deceleration, we applied a rotational equation of motion (\(\Delta \theta = \omega_0t + \frac{1}{2}\alpha t^2\) to calculate the angular displacement. This equation is analogous to the linear equation \(\Delta x = v_0t + \frac{1}{2}at^2\), where \(\Delta x\) is linear displacement and \(v_0\) and \(a\) are initial velocity and acceleration, respectively.

The calculated angular displacement (\(\Delta \theta\) then helps us determine how many full rotations the vinyl record made before coming to a halt. This exemplifies how the equations of motion are fundamental concept that bridges our understanding of motion, be it linear or rotational.