B. Suppose the wheel slows down uniformly, so that decreases by 8x rad/s every 4 s. (The wheel continues spinning in the same direction and has the same orientation.) Specify the angular acceleration a of the wheel by giving its magnitude and, relative to 0, its direction. In linear kinematics we find the acceleration vector by first constructing Av (a change in velocity vector) and then dividing that by At. Describe the analogous steps that you used above to find the angular acceleration a. -Discuss your answers above with a tutorial instructor before continuing.
B. Suppose the wheel slows down uniformly, so that decreases by 8x rad/s every 4 s. (The wheel continues spinning in the same direction and has the same orientation.) Specify the angular acceleration a of the wheel by giving its magnitude and, relative to 0, its direction. In linear kinematics we find the acceleration vector by first constructing Av (a change in velocity vector) and then dividing that by At. Describe the analogous steps that you used above to find the angular acceleration a. -Discuss your answers above with a tutorial instructor before continuing.
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Author:Raymond A. Serway, Chris Vuille
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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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![### Rotational Motion and Angular Velocity
#### Part I: Angular Displacement and Linear Speed
**F.** *In the space at right sketch the position vectors for point C at the beginning and at the end of a small time interval Δt.*
1. **Label the change in angle (Δθ) and the distance between the center of the wheel and point C (r_c). Sketch the path taken by point C during this time interval.**
**Sketch Explanation:**
- The sketch shows a circle representing the wheel, with the center marked.
- Two radial lines indicate the position vectors of point C at times t_i and t_i + Δt, separated by an angle Δθ.
- The distance between the center and point C is labeled as r_c.
**Mathematical Expression:**
- The distance that point C travels during Δt is given as:
\[ d = r_c \Delta \theta \]
2. **Use your answer above and the definition of linear speed to derive an algebraic expression for the linear speed of point C in terms of the angular speed ω of the wheel.**
**Derivation:**
- Linear speed \(v_c\) is defined as the distance traveled over the time interval Δt:
\[ v_c = \frac{d}{\Delta t} = \frac{r_c \Delta \theta}{\Delta t} \]
- Since angular speed \(ω\) is defined as:
\[ ω = \frac{\Delta \theta}{\Delta t} \]
- Substitute ω into the equation for \(v_c\):
\[ v_c = r_c \cdot ω \]
**Implication:**
- This implies that the linear speed is proportional to the distance from the center, meaning it is greater for points farther out on the wheel.
#### Part II: Motion with Changing Angular Velocity
**A.** *Let \( \vec{ω} \) represent the initial angular velocity of a wheel. In each case described below, determine the magnitude of the change in angular velocity \( \Delta \vec{ω} \) in terms of \( ω_0 \).*
1. **The wheel is made to spin faster, so that eventually, a fixed point on the wheel is going around twice as many times each second. (The axis of rotation is fixed.)**
**Calculation:**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F33f848af-7ea2-4b7e-a4c9-e1f33a68a6a6%2F3cad49ce-3e77-4438-8be5-27555d7285c3%2F5g20v4_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:### Rotational Motion and Angular Velocity
#### Part I: Angular Displacement and Linear Speed
**F.** *In the space at right sketch the position vectors for point C at the beginning and at the end of a small time interval Δt.*
1. **Label the change in angle (Δθ) and the distance between the center of the wheel and point C (r_c). Sketch the path taken by point C during this time interval.**
**Sketch Explanation:**
- The sketch shows a circle representing the wheel, with the center marked.
- Two radial lines indicate the position vectors of point C at times t_i and t_i + Δt, separated by an angle Δθ.
- The distance between the center and point C is labeled as r_c.
**Mathematical Expression:**
- The distance that point C travels during Δt is given as:
\[ d = r_c \Delta \theta \]
2. **Use your answer above and the definition of linear speed to derive an algebraic expression for the linear speed of point C in terms of the angular speed ω of the wheel.**
**Derivation:**
- Linear speed \(v_c\) is defined as the distance traveled over the time interval Δt:
\[ v_c = \frac{d}{\Delta t} = \frac{r_c \Delta \theta}{\Delta t} \]
- Since angular speed \(ω\) is defined as:
\[ ω = \frac{\Delta \theta}{\Delta t} \]
- Substitute ω into the equation for \(v_c\):
\[ v_c = r_c \cdot ω \]
**Implication:**
- This implies that the linear speed is proportional to the distance from the center, meaning it is greater for points farther out on the wheel.
#### Part II: Motion with Changing Angular Velocity
**A.** *Let \( \vec{ω} \) represent the initial angular velocity of a wheel. In each case described below, determine the magnitude of the change in angular velocity \( \Delta \vec{ω} \) in terms of \( ω_0 \).*
1. **The wheel is made to spin faster, so that eventually, a fixed point on the wheel is going around twice as many times each second. (The axis of rotation is fixed.)**
**Calculation:**
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