to a cord passing through a hole in the surface as in the figure. The puck is revo distance 2.0 m from the hole with an angular velocity of 3.0 rad/s. The cord is t from below, shortening the radius to 1.0 m. The new angular velocity (in rad/s)
to a cord passing through a hole in the surface as in the figure. The puck is revo distance 2.0 m from the hole with an angular velocity of 3.0 rad/s. The cord is t from below, shortening the radius to 1.0 m. The new angular velocity (in rad/s)
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Angular speed, acceleration and displacement
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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
![**Title: Angular Velocity of a Puck on a Frictionless Air Hockey Table**
**Introduction to Angular Momentum Conservation**
In this educational scenario, we explore the concept of angular momentum conservation using a puck on a frictionless air hockey table. A puck with a mass of 5.0 grams is tethered to a cord that threads through a hole in the tabletop. This setup allows the puck to rotate around the hole, shifting its dynamics when the cord's length is adjusted.
**Scenario Description**
Initially, the puck revolves at a radius of 2.0 meters from the hole with an angular velocity of 3.0 radians per second. Subsequently, the cord is shortened to a radius of 1.0 meter.
**Objective**
Determine the puck’s new angular velocity after the cord is pulled, altering the radius.
**Diagram Explanation**
The diagram illustrates:
- A flat air hockey table with a hole at the center.
- A puck attached to a cord moving in a circular path around the hole.
- The puck's motion is initially at a 2.0-meter radius with a 3.0 rad/s angular velocity.
- An arrow representing the direction for shortening the cord from below, reducing the radius to 1.0 meter.
**Key Concepts**
1. **Angular Momentum Conservation**: Since no external torque acts on the system, the initial and final angular momentum remain constant.
2. **Formula**:
\[
L = I \omega
\]
where \( L \) is angular momentum, \( I \) is moment of inertia, and \( \omega \) is angular velocity.
3. For a point mass:
\[
I = m \times r^2
\]
4. Initial and final angular momenta are equated:
\[
m \times (r_1)^2 \times \omega_1 = m \times (r_2)^2 \times \omega_2
\]
5. Solving for final angular velocity \( \omega_2 \):
\[
\omega_2 = \frac{(r_1)^2 \times \omega_1}{(r_2)^2}
\]
**Conclusion**
Using the illustrated scenario and these principles, we compute the puck's new angular velocity after the radius adjustment. This exercise underscores how changes in radius affect rotational speed, emphasizing angular momentum conservation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F877d147a-40f9-47d8-b25c-41d77d376d71%2F81649a19-e1ac-4559-bbb4-bbabfa4b319a%2F5gu4bt_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Angular Velocity of a Puck on a Frictionless Air Hockey Table**
**Introduction to Angular Momentum Conservation**
In this educational scenario, we explore the concept of angular momentum conservation using a puck on a frictionless air hockey table. A puck with a mass of 5.0 grams is tethered to a cord that threads through a hole in the tabletop. This setup allows the puck to rotate around the hole, shifting its dynamics when the cord's length is adjusted.
**Scenario Description**
Initially, the puck revolves at a radius of 2.0 meters from the hole with an angular velocity of 3.0 radians per second. Subsequently, the cord is shortened to a radius of 1.0 meter.
**Objective**
Determine the puck’s new angular velocity after the cord is pulled, altering the radius.
**Diagram Explanation**
The diagram illustrates:
- A flat air hockey table with a hole at the center.
- A puck attached to a cord moving in a circular path around the hole.
- The puck's motion is initially at a 2.0-meter radius with a 3.0 rad/s angular velocity.
- An arrow representing the direction for shortening the cord from below, reducing the radius to 1.0 meter.
**Key Concepts**
1. **Angular Momentum Conservation**: Since no external torque acts on the system, the initial and final angular momentum remain constant.
2. **Formula**:
\[
L = I \omega
\]
where \( L \) is angular momentum, \( I \) is moment of inertia, and \( \omega \) is angular velocity.
3. For a point mass:
\[
I = m \times r^2
\]
4. Initial and final angular momenta are equated:
\[
m \times (r_1)^2 \times \omega_1 = m \times (r_2)^2 \times \omega_2
\]
5. Solving for final angular velocity \( \omega_2 \):
\[
\omega_2 = \frac{(r_1)^2 \times \omega_1}{(r_2)^2}
\]
**Conclusion**
Using the illustrated scenario and these principles, we compute the puck's new angular velocity after the radius adjustment. This exercise underscores how changes in radius affect rotational speed, emphasizing angular momentum conservation.
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