M1, R1 M2, R2 The sphere on the right has 4 times the mass and 3 times the radius of the sphere on the left. What is the ratio of the density of the sphere on the right to the density of the sphere on the left? Give your answer in decimal form with two decimals.
M1, R1 M2, R2 The sphere on the right has 4 times the mass and 3 times the radius of the sphere on the left. What is the ratio of the density of the sphere on the right to the density of the sphere on the left? Give your answer in decimal form with two decimals.
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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![### Density Ratio Problem
#### Concept
Given are two spheres with mass and radius:
- Spheres on the left is represented with \( M_1 \) and \( R_1 \).
- Spheres on the right is represented with \( M_2 \) and \( R_2 \).
The problem statement provides the following information:
- The sphere on the right has 4 times the mass \( M \) and 3 times the radius \( R \) of the sphere on the left.
- You need to determine the ratio of the density of the sphere on the right to the density of the sphere on the left, providing your answer in decimal form with two decimal places.
#### Approach
The question involves the relationship between mass, radius, and density.
1. **Mass and Radius Relationship:**
\[ M_2 = 4 \times M_1 \]
\[ R_2 = 3 \times R_1 \]
2. **Density Relationship:**
Density (\( \rho \)) is defined as mass divided by volume. For a sphere, volume (\( V \)) is proportional to the cube of its radius.
\[ \rho = \frac{M}{V} \]
\[ V \propto R^3 \]
3. **Volume Calculation:**
\[ V_2 = \left(3 \times R_1\right)^3 = 27 \times R_1^3 \]
Since \( V_1 = R_1^3 \), therefore \( V_2 = 27 \times V_{1} \).
4. **Density Calculation:**
\[ \rho_1 = \frac{M_1}{V_1} \]
\[ \rho_2 = \frac{M_2}{V_2} = \frac{4 \times M_1}{27 \times V_1} = \frac{4}{27} \times \rho_1 \]
5. **Density Ratio:**
\[ \frac{\rho_2}{\rho_1} = \frac{4}{27} \approx 0.15 \]
Hence, the ratio of the density of the sphere on the right to the density of the sphere on the left is approximately \( \mathbf{0.15} \).
#### Summarized Result
**Ratio of Densities: 0.15**
This example illustrates how changing the radius and mass of a sphere affects](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffeee2f86-f729-492a-b5ec-fb1726086ce7%2F430f9a40-6571-49f8-a2d7-1ad3f0755e2c%2Fsdhp7i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Density Ratio Problem
#### Concept
Given are two spheres with mass and radius:
- Spheres on the left is represented with \( M_1 \) and \( R_1 \).
- Spheres on the right is represented with \( M_2 \) and \( R_2 \).
The problem statement provides the following information:
- The sphere on the right has 4 times the mass \( M \) and 3 times the radius \( R \) of the sphere on the left.
- You need to determine the ratio of the density of the sphere on the right to the density of the sphere on the left, providing your answer in decimal form with two decimal places.
#### Approach
The question involves the relationship between mass, radius, and density.
1. **Mass and Radius Relationship:**
\[ M_2 = 4 \times M_1 \]
\[ R_2 = 3 \times R_1 \]
2. **Density Relationship:**
Density (\( \rho \)) is defined as mass divided by volume. For a sphere, volume (\( V \)) is proportional to the cube of its radius.
\[ \rho = \frac{M}{V} \]
\[ V \propto R^3 \]
3. **Volume Calculation:**
\[ V_2 = \left(3 \times R_1\right)^3 = 27 \times R_1^3 \]
Since \( V_1 = R_1^3 \), therefore \( V_2 = 27 \times V_{1} \).
4. **Density Calculation:**
\[ \rho_1 = \frac{M_1}{V_1} \]
\[ \rho_2 = \frac{M_2}{V_2} = \frac{4 \times M_1}{27 \times V_1} = \frac{4}{27} \times \rho_1 \]
5. **Density Ratio:**
\[ \frac{\rho_2}{\rho_1} = \frac{4}{27} \approx 0.15 \]
Hence, the ratio of the density of the sphere on the right to the density of the sphere on the left is approximately \( \mathbf{0.15} \).
#### Summarized Result
**Ratio of Densities: 0.15**
This example illustrates how changing the radius and mass of a sphere affects
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