**Problem Statement:**A turntable starts from 10 rad/s of angular velocity and slows down to a stop in 5.0 s. Find the total angle turned during this time, in rad.**Hint:** Use rotational kinematics formula. Assume initial angle and final angular velocity as zero.---**Solution:**1. **Given Data:** - Initial angular velocity (\(\omega_0\)): 10 rad/s - Final angular velocity (\(\omega\)): 0 rad/s (since it comes to a stop) - Time interval (t): 5.0 s - Initial angle (\(\theta_0\)): 0 rad2. **Find:** Total angle turned (\(\Delta \theta\))3. **Rotational Kinematics Formula:** The formula for angular displacement with constant angular acceleration is: \[ \theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2 \] Where: - \(\theta\) is the final angular position - \(\theta_0\) is the initial angular position - \(\omega_0\) is the initial angular velocity - \(t\) is the time - \(\alpha\) is the angular acceleration4. **Calculate Angular Acceleration (\(\alpha\)):** Using the formula: \[ \omega = \omega_0 + \alpha t \] Since the final angular velocity (\(\omega\)) is 0: \[ 0 = 10 + \alpha (5) \] \[ \alpha = -2 \ \text{rad/s}^2 \]5. **Calculate Total Angle Turned (\(\Delta \theta\)):** Using the kinematic formula: \[ \theta = 0 + (10 \ \text{rad/s}) (5 \ \text{s}) + \frac{1}{2} (-2 \ \text{rad/s}^2) (5 \ \text{s})^2 \] \[ \theta = 10 (5) + \frac{1}{2} (-2) (25) \] \[ \theta = 50 - 25 \] \

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A turn table starts from 10 rad/s of angular velocity and slows down to a stop in 5.0 s. Find the total angle turned during this time, in rad.

College Physics

1st Edition

ISBN:9781938168000

Author:Paul Peter Urone, Roger Hinrichs

Publisher:Paul Peter Urone, Roger Hinrichs

Chapter10: Rotational Motion And Angular Momentum

Section: Chapter Questions

Problem 21PE: This problem considers energy and work aspects of Example 10.7—use data from that example as needed....

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Concept explainers

Angular speed, acceleration and displacement

Angular acceleration is defined as the rate of change in angular velocity with respect to time. It has both magnitude and direction. So, it is a vector quantity.

Angular Position

Before diving into angular position, one should understand the basics of position and its importance along with usage in day-to-day life. When one talks of position, it’s always relative with respect to some other object. For example, position of earth with respect to sun, position of school with respect to house, etc. Angular position is the rotational analogue of linear position.

Question

**Problem Statement:**

A turntable starts from 10 rad/s of angular velocity and slows down to a stop in 5.0 s. Find the total angle turned during this time, in rad.

**Hint:** Use rotational kinematics formula. Assume initial angle and final angular velocity as zero.

---

**Solution:**

1. **Given Data:**
- Initial angular velocity (\(\omega_0\)): 10 rad/s
- Final angular velocity (\(\omega\)): 0 rad/s (since it comes to a stop)
- Time interval (t): 5.0 s
- Initial angle (\(\theta_0\)): 0 rad

2. **Find:** Total angle turned (\(\Delta \theta\))

3. **Rotational Kinematics Formula:**
The formula for angular displacement with constant angular acceleration is:
\[
\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2
\]

Where:
- \(\theta\) is the final angular position
- \(\theta_0\) is the initial angular position
- \(\omega_0\) is the initial angular velocity
- \(t\) is the time
- \(\alpha\) is the angular acceleration

4. **Calculate Angular Acceleration (\(\alpha\)):**
Using the formula:
\[
\omega = \omega_0 + \alpha t
\]

Since the final angular velocity (\(\omega\)) is 0:
\[
0 = 10 + \alpha (5)
\]
\[
\alpha = -2 \ \text{rad/s}^2
\]

5. **Calculate Total Angle Turned (\(\Delta \theta\)):**
Using the kinematic formula:
\[
\theta = 0 + (10 \ \text{rad/s}) (5 \ \text{s}) + \frac{1}{2} (-2 \ \text{rad/s}^2) (5 \ \text{s})^2
\]
\[
\theta = 10 (5) + \frac{1}{2} (-2) (25)
\]
\[
\theta = 50 - 25
\]
\

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