Part C Calculate the instantaneous value of the angular velocity wz at t= 5.05 s. 1D ΑΣΦ www ? rad/s Request Answer Part D Calculate the average angular velocity wav for the time interval t=0 to £= 5.05 s. VD ΑΣΦ ? Way = rad/s Submit Request Answer لیا Submit
Part C Calculate the instantaneous value of the angular velocity wz at t= 5.05 s. 1D ΑΣΦ www ? rad/s Request Answer Part D Calculate the average angular velocity wav for the time interval t=0 to £= 5.05 s. VD ΑΣΦ ? Way = rad/s Submit Request Answer لیا Submit
Classical Dynamics of Particles and Systems
5th Edition
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Stephen T. Thornton, Jerry B. Marion
Chapter8: Central-force Motion
Section: Chapter Questions
Problem 8.9P
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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
9.5
1. Calculate the angular velocity of the merry-go-round as a function of time.
Express your answer in terms of the variables β, γ, and t.
2. What is the initial value of the angular velocity?
other 2 questions are attached
![### Part C
#### Calculate the instantaneous value of the angular velocity \(\omega_z\) at \( t = 5.05 \, \text{s} \).
##### Input Box
\[ \omega_z = \quad \quad \quad \quad \quad \quad \quad \text{rad/s} \]
**Buttons Available for Input:**
- **Graph (Function) Tool**: Allows for graphing functions and equations.
- **Equation Tool**: Insertion of mathematical equations.
- **Undo/Redo Buttons**: To undo or redo recent changes.
- **Statistics Tool**: For statistical calculations.
- **Greek Symbols Tool**: For inserting Greek letters.
- **Reset Button**: Clears the entered data.
- **Constants Button**: For inserting physical constants.
- **Help Button**: Provides additional help.
**Action Buttons:**
- **Submit**: Submits your calculated answer.
- **Request Answer**: Requests the correct answer.
### Part D
#### Calculate the average angular velocity \(\omega_{\text{av}_z}\) for the time interval \( t = 0 \) to \( t = 5.05 \, \text{s} \).
##### Input Box
\[ \omega_{\text{av}_z} = \quad \quad \quad \quad \quad \quad \quad \text{rad/s} \]
**Buttons Available for Input:**
- **Graph (Function) Tool**: Allows for graphing functions and equations.
- **Equation Tool**: Insertion of mathematical equations.
- **Undo/Redo Buttons**: To undo or redo recent changes.
- **Statistics Tool**: For statistical calculations.
- **Greek Symbols Tool**: For inserting Greek letters.
- **Reset Button**: Clears the entered data.
- **Constants Button**: For inserting physical constants.
- **Help Button**: Provides additional help.
**Action Buttons:**
- **Submit**: Submits your calculated answer.
- **Request Answer**: Requests the correct answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fddf14729-dbf2-4574-b793-bebd8df9c378%2Febdb9105-2c68-4888-b560-8a81ba73fbd9%2Fdr01znq_processed.png&w=3840&q=75)
Transcribed Image Text:### Part C
#### Calculate the instantaneous value of the angular velocity \(\omega_z\) at \( t = 5.05 \, \text{s} \).
##### Input Box
\[ \omega_z = \quad \quad \quad \quad \quad \quad \quad \text{rad/s} \]
**Buttons Available for Input:**
- **Graph (Function) Tool**: Allows for graphing functions and equations.
- **Equation Tool**: Insertion of mathematical equations.
- **Undo/Redo Buttons**: To undo or redo recent changes.
- **Statistics Tool**: For statistical calculations.
- **Greek Symbols Tool**: For inserting Greek letters.
- **Reset Button**: Clears the entered data.
- **Constants Button**: For inserting physical constants.
- **Help Button**: Provides additional help.
**Action Buttons:**
- **Submit**: Submits your calculated answer.
- **Request Answer**: Requests the correct answer.
### Part D
#### Calculate the average angular velocity \(\omega_{\text{av}_z}\) for the time interval \( t = 0 \) to \( t = 5.05 \, \text{s} \).
##### Input Box
\[ \omega_{\text{av}_z} = \quad \quad \quad \quad \quad \quad \quad \text{rad/s} \]
**Buttons Available for Input:**
- **Graph (Function) Tool**: Allows for graphing functions and equations.
- **Equation Tool**: Insertion of mathematical equations.
- **Undo/Redo Buttons**: To undo or redo recent changes.
- **Statistics Tool**: For statistical calculations.
- **Greek Symbols Tool**: For inserting Greek letters.
- **Reset Button**: Clears the entered data.
- **Constants Button**: For inserting physical constants.
- **Help Button**: Provides additional help.
**Action Buttons:**
- **Submit**: Submits your calculated answer.
- **Request Answer**: Requests the correct answer.
![**Angular Motion of a Merry-Go-Round**
A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to the equation:
\[
\theta(t) = \gamma t + \beta t^3
\]
where:
- \(\gamma = 0.406 \, \text{rad/s}\)
- \(\beta = 1.30 \times 10^{-2} \, \text{rad/s}^3\)
In this equation:
- \(\theta(t)\) represents the angle in radians through which the merry-go-round has turned after time \(t\).
- \(\gamma\) is a constant representing the angular velocity component in radians per second.
- \(\beta\) is a constant representing the angular acceleration component that influences the cubic term of time in the equation.
This mathematical model helps to describe the rotational motion of the merry-go-round over time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fddf14729-dbf2-4574-b793-bebd8df9c378%2Febdb9105-2c68-4888-b560-8a81ba73fbd9%2F6viat1d_processed.png&w=3840&q=75)
Transcribed Image Text:**Angular Motion of a Merry-Go-Round**
A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to the equation:
\[
\theta(t) = \gamma t + \beta t^3
\]
where:
- \(\gamma = 0.406 \, \text{rad/s}\)
- \(\beta = 1.30 \times 10^{-2} \, \text{rad/s}^3\)
In this equation:
- \(\theta(t)\) represents the angle in radians through which the merry-go-round has turned after time \(t\).
- \(\gamma\) is a constant representing the angular velocity component in radians per second.
- \(\beta\) is a constant representing the angular acceleration component that influences the cubic term of time in the equation.
This mathematical model helps to describe the rotational motion of the merry-go-round over time.
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