Problem 12 - College Physics

Question

\(\cdot\) When the power is turned off on a turntable spinning at 78.0 rpm, you find that it takes 10.5 revolutions for it to stop while slowing down at a uniform rate. (a) What is the angular acceleration (in rad \(s^{2} )\) of this turntable? (b) How long does it take to stop after the power is turned off?

Step-by-Step Solution

Answer

(a) The angular acceleration is approximately -0.505 rad/s². (b) It takes about 16.18 seconds to stop.

Step 1: Convert RPM to Radians per Second

First, convert the initial angular speed from rotations per minute (rpm) to radians per second. We know that 1 revolution is equal to \(2\pi\) radians and there are 60 seconds in a minute. So:\[\omega_i = 78.0 \text{ rpm} \times \frac{2\pi \text{ rad}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}} = \frac{78.0 \times 2\pi}{60}\, \text{rad/s} \approx 8.168 \text{ rad/s}\]

Step 2: Calculate Total Angular Displacement

Next, convert the angular displacement from revolutions to radians. We know it makes 10.5 revolutions before stopping:\[\theta = 10.5 \text{ revolutions} \times 2\pi \text{ rad/revolution} = 21\pi \text{ radians}\]

Step 3: Use Kinematic Equation for Angular Motion

We use the kinematic equation for angular motion to find the angular acceleration, \(\alpha\) (rad/s²), assuming the final angular velocity, \(\omega_f = 0\) as the turntable stops:\[\omega_f^2 = \omega_i^2 + 2\alpha\theta\]Rearranging gives:\[0 = (8.168)^2 + 2\alpha(21\pi)\]\[ \alpha = -\frac{(8.168)^2}{2 \times 21\pi} \approx -0.505 \text{ rad/s}^2 \]

Step 4: Calculate Time to Stop

We use another kinematic equation for motion to find the time, \(t\), it takes for the turntable to stop:\[\omega_f = \omega_i + \alpha t\]Rearranging gives:\[0 = 8.168 + (-0.505) t\]\[ t = \frac{8.168}{0.505} \approx 16.18 \text{ seconds}\]

Key Concepts

Angular Acceleration

Angular acceleration is a measure of how quickly the angular velocity of an object changes. Just like linear acceleration is measured in meters per second squared, angular acceleration is measured in radians per second squared (rad/s²). When a rotating object, like the turntable in our exercise, slows down, it experiences a negative angular acceleration. This negative value indicates that the rotation is decelerating.
To find the angular acceleration for our turntable, we use one of the kinematic equations. In this case, we derive angular acceleration from the initial angular velocity and the angular displacement, taking into account that the final angular velocity is zero because the turntable comes to a stop.

Kinematic Equations

Kinematic equations are crucial in solving problems related to motion, whether linear or angular. For angular motion, these equations relate angular displacement, angular velocity, angular acceleration, and time.
Specifically, the kinematic equation used in our problem is \[\omega_f^2 = \omega_i^2 + 2\alpha\theta\]where:

• \(\omega_f\) is the final angular velocity.
• \(\omega_i\) is the initial angular velocity.
• \(\alpha\) is the angular acceleration.
• \(\theta\) is the angular displacement.

By rearranging this equation, we derive the angular acceleration, vital for understanding how quickly a rotating system can increase or decrease its speed.

Revolutions per Minute

Revolutions per minute (rpm) is a common unit of angular velocity, describing how many full rotations an object completes in one minute. It's commonly used in settings like turntables, motors, and engines.
To work with angular motion in standard units, such as radians per second, it is often necessary to convert rpm to rad/s. This conversion involves multiplying by the factor \(\frac{2\pi}{60}\), converting the number of rotations into radians and the time unit from minutes to seconds. In this exercise, the turntable's speed of 78 rpm translates to approximately 8.168 rad/s, which lays the groundwork for further calculations.

Angular Displacement

Angular displacement measures the angle through which an object rotates, in radians, over a given time span. Unlike linear displacement, which is measured in meters, angular displacement captures how far an object has turned.
In our problem, calculating angular displacement involves converting 10.5 revolutions into radians. Since one revolution is equal to \(2\pi\) radians, multiplying the number of revolutions by this factor gives us the total angular displacement (21\(\pi\) radians) needed for using kinematic equations.

Radians per Second

Radians per second (rad/s) is a standard unit of angular velocity, showing how fast an object rotates about its axis. It is the angular equivalent of meters per second in linear motion.
This unit is particularly useful for applications where angular velocity needs to work seamlessly with angular displacement and acceleration, as seen in our exercise. Through conversion, we ensure a coherent system of units, essential for accurate calculation and comprehension of angular dynamics.