You are to find the coordinates of a pebble stuck i n the tread of a rolling tire that is rotating counterclockwise (i.e., in the positive sense) with angular velocity w. The tire rolls without slipping on the ground (which is at y=0). The outer radius of the tire is R. At time t=0 the pebble is at the top of the tire, as shown. (Figure 1) Figure 1=0 ros < 1 of 2 > Part D Now find ape (t), the acceleration vector of the pebble with respect to a fixed point on the ground. Express your answer in terms of R, w, t and i and/or of the xy coordinate system shown. View Available Hint(s) aps (t)= Rw²sin (wt) i- Rw²cos (wt) 3 Submit ✓ Correct Part E Previous Answers Now find the magnitude of the acceleration vector. Your answer should be independent of time. ▸ View Available Hint(s) a(t) = |apg(t) = Submit Provide Feedback IVE ΑΣΦ ?
You are to find the coordinates of a pebble stuck i n the tread of a rolling tire that is rotating counterclockwise (i.e., in the positive sense) with angular velocity w. The tire rolls without slipping on the ground (which is at y=0). The outer radius of the tire is R. At time t=0 the pebble is at the top of the tire, as shown. (Figure 1) Figure 1=0 ros < 1 of 2 > Part D Now find ape (t), the acceleration vector of the pebble with respect to a fixed point on the ground. Express your answer in terms of R, w, t and i and/or of the xy coordinate system shown. View Available Hint(s) aps (t)= Rw²sin (wt) i- Rw²cos (wt) 3 Submit ✓ Correct Part E Previous Answers Now find the magnitude of the acceleration vector. Your answer should be independent of time. ▸ View Available Hint(s) a(t) = |apg(t) = Submit Provide Feedback IVE ΑΣΦ ?
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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![### Rolling Tire Pebble Coordinates - Educational Module
#### Problem Context
You are tasked with finding the coordinates of a pebble embedded in the tread of a tire that is rotating counterclockwise (in the positive sense) with an angular velocity, \( \omega \). The tire rolls without slipping on the ground (which is at \( y = 0 \)). The outer radius of the tire is \( R \). Initially, at \( t = 0 \), the pebble is located at the top of the tire, as depicted.
#### Figure Description
- The diagram shows a tire with a radius \( R \) and a center point. The tire is rotating counterclockwise.
- A coordinate system with axes \( x \) and \( y \) is depicted with the origin at the center of the tire.
- At time \( t = 0 \), the pebble is shown at the top of the tire.
#### Parts of the Problem
**Part D**
- **Task:** Determine the acceleration vector \( \vec{a}_{\text{peb}}(t) \) of the pebble with respect to a fixed point on the ground.
- **Solution Formula:**
\[
\vec{a}_{\text{peb}}(t) = -R\omega^2 \sin(\omega t) \hat{i} - R\omega^2 \cos(\omega t) \hat{j}
\]
- **Hint Provided:** Encourage use of the given formula in terms of \( R, \omega, t \) and unit vectors \( \hat{i} \) and \( \hat{j} \).
- *Submission Status:* Correct
**Part E**
- **Task:** Find the magnitude of the acceleration vector.
- **Requirement:** The answer should be independent of time.
- **Formula Entry Area:** There is a field to input the magnitude of the acceleration vector \( a(t) = |\vec{a}_{\text{peb}}(t)| \).
#### Copyright Information
Content provided by Pearson Education Inc. All rights reserved.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb2fd9b27-3894-4c03-89ce-047a1b838456%2F30fc36dc-838b-44da-914e-4ea5a535138a%2Flaqa9pd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Rolling Tire Pebble Coordinates - Educational Module
#### Problem Context
You are tasked with finding the coordinates of a pebble embedded in the tread of a tire that is rotating counterclockwise (in the positive sense) with an angular velocity, \( \omega \). The tire rolls without slipping on the ground (which is at \( y = 0 \)). The outer radius of the tire is \( R \). Initially, at \( t = 0 \), the pebble is located at the top of the tire, as depicted.
#### Figure Description
- The diagram shows a tire with a radius \( R \) and a center point. The tire is rotating counterclockwise.
- A coordinate system with axes \( x \) and \( y \) is depicted with the origin at the center of the tire.
- At time \( t = 0 \), the pebble is shown at the top of the tire.
#### Parts of the Problem
**Part D**
- **Task:** Determine the acceleration vector \( \vec{a}_{\text{peb}}(t) \) of the pebble with respect to a fixed point on the ground.
- **Solution Formula:**
\[
\vec{a}_{\text{peb}}(t) = -R\omega^2 \sin(\omega t) \hat{i} - R\omega^2 \cos(\omega t) \hat{j}
\]
- **Hint Provided:** Encourage use of the given formula in terms of \( R, \omega, t \) and unit vectors \( \hat{i} \) and \( \hat{j} \).
- *Submission Status:* Correct
**Part E**
- **Task:** Find the magnitude of the acceleration vector.
- **Requirement:** The answer should be independent of time.
- **Formula Entry Area:** There is a field to input the magnitude of the acceleration vector \( a(t) = |\vec{a}_{\text{peb}}(t)| \).
#### Copyright Information
Content provided by Pearson Education Inc. All rights reserved.
![### Educational Content: Acceleration in Rotational Motion
#### Problem Statement
**Part E:**
You are asked to find the magnitude of the acceleration vector. Note that your answer should be independent of time.
#### Diagram Explanation
The diagram features a large circle representing a wheel, with a smaller point mass \( P \) located on its rim. The wheel rotates about its center. Key elements include:
- **\( \mathbf{r}_{\text{rel}} \):** The position vector from the center of the wheel to the point mass \( P \).
- **\( \omega \):** The angular velocity vector pointing upwards, perpendicular to the wheel's plane.
- **\( R \):** The radius of the wheel.
- **x and y Axes:** The diagram is oriented with horizontal (x) and vertical (y) axes for reference.
#### Solution Interface
A text box is provided for entering the solution. Relevant mathematical symbols can be inserted using the built-in editor.
#### Example Input Formula
The formula for the acceleration in terms of the radius and angular velocity is given as:
\[ \vec{a}_{\text{rel}}(t) = -R \omega^2 \sin(\omega t) \, \hat{\imath} - R \omega^2 \cos(\omega t) \, \hat{\jmath} \]
Please submit your answer through the interface provided.
---
**Disclaimer:** This is educational content provided by Pearson Education Inc. and adheres to their trademark and privacy policies.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb2fd9b27-3894-4c03-89ce-047a1b838456%2F30fc36dc-838b-44da-914e-4ea5a535138a%2Fdk58uks_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Content: Acceleration in Rotational Motion
#### Problem Statement
**Part E:**
You are asked to find the magnitude of the acceleration vector. Note that your answer should be independent of time.
#### Diagram Explanation
The diagram features a large circle representing a wheel, with a smaller point mass \( P \) located on its rim. The wheel rotates about its center. Key elements include:
- **\( \mathbf{r}_{\text{rel}} \):** The position vector from the center of the wheel to the point mass \( P \).
- **\( \omega \):** The angular velocity vector pointing upwards, perpendicular to the wheel's plane.
- **\( R \):** The radius of the wheel.
- **x and y Axes:** The diagram is oriented with horizontal (x) and vertical (y) axes for reference.
#### Solution Interface
A text box is provided for entering the solution. Relevant mathematical symbols can be inserted using the built-in editor.
#### Example Input Formula
The formula for the acceleration in terms of the radius and angular velocity is given as:
\[ \vec{a}_{\text{rel}}(t) = -R \omega^2 \sin(\omega t) \, \hat{\imath} - R \omega^2 \cos(\omega t) \, \hat{\jmath} \]
Please submit your answer through the interface provided.
---
**Disclaimer:** This is educational content provided by Pearson Education Inc. and adheres to their trademark and privacy policies.
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