**Problem Statement:**The flywheel of a steam engine begins to rotate from rest with a constant angular acceleration of 1.23 rad/s². It accelerates for 31.9 seconds, then maintains a constant angular velocity. Calculate the total angle through which the wheel has turned 65.5 seconds after it begins rotating.**Solution Approach:**1. **Calculate Angular Displacement During Acceleration Phase:** - Use the formula for angular displacement during uniform acceleration: \[ \theta_1 = \omega_0 t + \frac{1}{2} \alpha t^2 \] - Where \(\omega_0 = 0\) (initial angular velocity), \(\alpha = 1.23 \, \text{rad/s}^2\), and \(t = 31.9 \, \text{s}\).2. **Calculate Angular Velocity at the End of Acceleration Phase:** - Angular velocity (\(\omega\)) can be calculated using: \[ \omega = \omega_0 + \alpha t \]3. **Calculate Angular Displacement During Constant Velocity Phase:** - After 31.9 seconds, the wheel travels with constant angular velocity for \(65.5 - 31.9 = 33.6 \, \text{s}\). - Angular displacement during this phase (\(\theta_2\)): \[ \theta_2 = \omega \times 33.6 \, \text{s} \]4. **Calculate Total Angular Displacement:** - Total angle \(\theta\) is the sum of \(\theta_1\) and \(\theta_2\): \[ \theta = \theta_1 + \theta_2 \]**Visual Aid:**- No graphs or diagrams are present in the original problem statement. A detailed step-by-step explanation accompanies the problem for clarity.

Given data: Initial angular velocity (ω0) = 0 rad/s Constant angular acceleration (α) = 1.23 rad/s2…

The flywheel of a steam engine begins to rotate from rest with a constant angular acceleration of 1.23 rad/s². It accelerates for 31.9 s, then maintains a constant angular velocity. Calculate the total angle through which the wheel has turned 65.5 s after it begins rotating.

The flywheel of a steam engine begins to rotate from rest with a constant angular acceleration of 1.23 rad/s². It accelerates for 31.9 s, then maintains a constant angular velocity. Calculate the total angle through which the wheel has turned 65.5 s after it begins rotating.

Physics for Scientists and Engineers: Foundations and Connections

1st Edition

ISBN:9781133939146

Author:Katz, Debora M.

Publisher:Katz, Debora M.

Chapter12: Rotation I: Kinematics And Dynamics

Section: Chapter Questions

Problem 63PQ

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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:**

The flywheel of a steam engine begins to rotate from rest with a constant angular acceleration of 1.23 rad/s². It accelerates for 31.9 seconds, then maintains a constant angular velocity. Calculate the total angle through which the wheel has turned 65.5 seconds after it begins rotating.

**Solution Approach:**

1. **Calculate Angular Displacement During Acceleration Phase:**
- Use the formula for angular displacement during uniform acceleration:
\[
\theta_1 = \omega_0 t + \frac{1}{2} \alpha t^2
\]
- Where \(\omega_0 = 0\) (initial angular velocity), \(\alpha = 1.23 \, \text{rad/s}^2\), and \(t = 31.9 \, \text{s}\).

2. **Calculate Angular Velocity at the End of Acceleration Phase:**
- Angular velocity (\(\omega\)) can be calculated using:
\[
\omega = \omega_0 + \alpha t
\]

3. **Calculate Angular Displacement During Constant Velocity Phase:**
- After 31.9 seconds, the wheel travels with constant angular velocity for \(65.5 - 31.9 = 33.6 \, \text{s}\).
- Angular displacement during this phase (\(\theta_2\)):
\[
\theta_2 = \omega \times 33.6 \, \text{s}
\]

4. **Calculate Total Angular Displacement:**
- Total angle \(\theta\) is the sum of \(\theta_1\) and \(\theta_2\):
\[
\theta = \theta_1 + \theta_2
\]

**Visual Aid:**

- No graphs or diagrams are present in the original problem statement. A detailed step-by-step explanation accompanies the problem for clarity.

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