A wheel with a radius of 1 meter is initially spinning with an angular velocity of 10 rotations minute. A break is applied to the edge of the wheel which causes it to slow the rate of its spin with constant acceleration. per a. If the rate of angular acceleration is -5 rad/s², how many seconds pass between the time the break is applied, and the wheel turns another 720 degrees?
A wheel with a radius of 1 meter is initially spinning with an angular velocity of 10 rotations minute. A break is applied to the edge of the wheel which causes it to slow the rate of its spin with constant acceleration. per a. If the rate of angular acceleration is -5 rad/s², how many seconds pass between the time the break is applied, and the wheel turns another 720 degrees?
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The wheel is spinning at an initial
Transcribed Image Text:**Title: Understanding Rotational Motion and Angular Deceleration**
**Introduction:**
This educational resource explores the principles of rotational motion and angular deceleration through a practical problem involving a spinning wheel. The exercise illustrates how to calculate the time taken for a wheel to decelerate under constant angular acceleration and how to determine centripetal acceleration at a specific moment.
**Problem Description:**
Consider a wheel with the following characteristics:
- Radius: 1 meter
- Initial Angular Velocity: 10 rotations per minute
A braking force is applied to the edge of the wheel, resulting in a constant deceleration.
**Questions:**
a. Given an angular acceleration of \(-5 \, \text{rad/s}^2\), find the time it takes for the wheel to turn an additional 720 degrees after the brake is applied.
b. Determine the centripetal (radial) acceleration of a point on the edge of the wheel at the instant it completes this additional 720-degree rotation.
**Solution Overview:**
1. **Convert Initial Velocity:**
- 10 rotations per minute = \(\frac{10 \times 2\pi}{60} \, \text{rad/s}\).
2. **Angular Displacement:**
- 720 degrees = \(4\pi \, \text{radians}\).
3. **Using Kinematic Equations for Rotation:**
- Use \(\theta = \omega_0 t + \frac{1}{2}\alpha t^2\) to solve for \(t\).
4. **Centripetal Acceleration:**
- Use \(a_c = \omega^2 r\) to find the radial acceleration once \(\omega\) is determined at 720 degrees rotation.
By exploring these calculations, students gain a deeper understanding of rotational dynamics, angular deceleration, and related concepts. This exercise is designed to bridge theoretical knowledge with practical application, enhancing learning outcomes.
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