The figure shows a uniform 66-g wire bent into a right triangle with L = 13.1 cm and H = 8.5 cm. Calculate the distance rem from the bottom left corner to the wire's center of mass. Report your answer in cm. [Hint: Each segment's mass is proportional to its length divided by the wire's total length. Also, the center of the hypotenuse is halfway across the horizontal leg and halfway up the vertical leg.]
The figure shows a uniform 66-g wire bent into a right triangle with L = 13.1 cm and H = 8.5 cm. Calculate the distance rem from the bottom left corner to the wire's center of mass. Report your answer in cm. [Hint: Each segment's mass is proportional to its length divided by the wire's total length. Also, the center of the hypotenuse is halfway across the horizontal leg and halfway up the vertical leg.]
College Physics
11th Edition
ISBN:9781305952300
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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![### Educational Content: Calculating the Center of Mass of a Bent Wire
#### Problem Statement
The figure shows a uniform 66-g wire bent into a right triangle with \(L = 13.1 \, \text{cm}\) and \(H = 8.5 \, \text{cm}\). Calculate the distance \(r_{cm}\) from the bottom left corner to the wire's center of mass. Report your answer in cm.
**Hint:** Each segment's mass is proportional to its length divided by the wire’s total length. Also, the center of the hypotenuse is halfway across the horizontal leg and halfway up the vertical leg.
#### Diagram Information
A right triangle is depicted with:
- Base \(L\) measuring \(13.1 \, \text{cm}\).
- Height \(H\) measuring \(8.5 \, \text{cm}\).
The dimensions create a right-angled triangle where the wire follows the triangle’s perimeter.
#### Task
1. **Determine Segment Lengths:**
- Calculate the hypotenuse using the Pythagorean theorem:
\[
\text{Hypotenuse} = \sqrt{L^2 + H^2}
\]
2. **Calculate the Mass Proportion:**
- Determine the length of each segment.
- Use the ratio of segment length to the total wire length to find the mass distribution.
3. **Find the Center of Mass Coordinates:**
- Calculate the x and y coordinates of the wire's center of mass using:
\[
r_{cm} = \frac{\sum (m_i \cdot r_i)}{M}
\]
- Sum the moments about each axis for the segments.
4. **Result:**
- Record your numerical answer assuming three significant figures.
#### Answer Box
Use the space provided below to enter the calculated center of mass:
\[ r_{cm} = \, \_\_\_\_ \, \text{cm} \]
> **Note:** This exercise aids in understanding the calculation of the center of mass for composite objects by simplifying the problem into segments of uniform mass distribution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F047a7e6a-f025-4b5b-ab83-4ffe14f69253%2F460ed8ee-327f-4bbd-b9ed-8da155c9058a%2Fnm4172p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Content: Calculating the Center of Mass of a Bent Wire
#### Problem Statement
The figure shows a uniform 66-g wire bent into a right triangle with \(L = 13.1 \, \text{cm}\) and \(H = 8.5 \, \text{cm}\). Calculate the distance \(r_{cm}\) from the bottom left corner to the wire's center of mass. Report your answer in cm.
**Hint:** Each segment's mass is proportional to its length divided by the wire’s total length. Also, the center of the hypotenuse is halfway across the horizontal leg and halfway up the vertical leg.
#### Diagram Information
A right triangle is depicted with:
- Base \(L\) measuring \(13.1 \, \text{cm}\).
- Height \(H\) measuring \(8.5 \, \text{cm}\).
The dimensions create a right-angled triangle where the wire follows the triangle’s perimeter.
#### Task
1. **Determine Segment Lengths:**
- Calculate the hypotenuse using the Pythagorean theorem:
\[
\text{Hypotenuse} = \sqrt{L^2 + H^2}
\]
2. **Calculate the Mass Proportion:**
- Determine the length of each segment.
- Use the ratio of segment length to the total wire length to find the mass distribution.
3. **Find the Center of Mass Coordinates:**
- Calculate the x and y coordinates of the wire's center of mass using:
\[
r_{cm} = \frac{\sum (m_i \cdot r_i)}{M}
\]
- Sum the moments about each axis for the segments.
4. **Result:**
- Record your numerical answer assuming three significant figures.
#### Answer Box
Use the space provided below to enter the calculated center of mass:
\[ r_{cm} = \, \_\_\_\_ \, \text{cm} \]
> **Note:** This exercise aids in understanding the calculation of the center of mass for composite objects by simplifying the problem into segments of uniform mass distribution.
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