A block of mass 4.9 kg is sitting on a frictionless ramp with a spring at the bottom that has a spring constant of 410 N/m (refer to the figure). The angle of the ramp with respect to the horizontal is 19°.1.The block, starting from rest, slides down the ramp a distance 32 cm before hitting the spring. How far, in centimeters, is the spring compressed as the block comes to momentary rest? 2.After the block comes to rest, the spring pushes the block back up the ramp. How fast, in meters per second, is the block moving right after it comes off the spring? 3.What is the change of the gravitational potential energy, in joules, between the original position of the block at the top of the ramp and the position of the block when the spring is fully compressed? ### Transcription for Educational Website#### Physics Diagram Explanation: Spring-Mass System on an Inclined PlaneThe diagram illustrates a spring-mass system situated on an inclined plane. The key components and their notations are as follows:1. **Mass (M)**: The mass, denoted by the symbol "M," represents the object placed on the inclined plane. It is depicted in orange.2. **Inclined Plane**: The plane on which the mass rests is inclined at an angle, denoted by the Greek letter theta (θ).3. **Spring**: A spring is attached to one end of the inclined plane and is connected to the mass (M). 4. **Distance (d)**: The distance "d" is the length between the equilibrium position of the spring and the current position of the mass.5. **Length (l)**: The length "l" refers to the length of the spring when it is neither compressed nor stretched.6. **Inclination Angle (θ)**: The angle of inclination of the plane with respect to the horizontal surface is denoted by theta (θ).This diagram is useful in studying the concepts of mechanical energy, forces, and motion on an inclined plane, specifically:- **Potential Energy in the Spring**: Given by the formula \( \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.- **Gravitational Potential Energy**: The potential energy due to the height of the mass on the inclined plane.- **Kinetic Energy**: If the mass is in motion, its kinetic energy would also be considered.- **Forces at Play**: Analysis includes gravitational force (Fg), normal force perpendicular to the plane's surface, and the restoring force exerted by the spring.This information is critical in understanding the dynamics involved in spring-mass systems on inclined planes, which is a fundamental concept in classical mechanics.

mass of block (m) = 4.9 kg spring constant (k) = 410 N/m θ = 19o

Answered: A block of mass 4.9 kg is sitting on a…

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ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

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A block of mass 4.9 kg is sitting on a frictionless ramp with a spring at the bottom that has a spring constant of 410 N/m (refer to the figure). The angle of the ramp with respect to the horizontal is 19°.

1.The block, starting from rest, slides down the ramp a distance 32 cm before hitting the spring. How far, in centimeters, is the spring compressed as the block comes to momentary rest?

2.After the block comes to rest, the spring pushes the block back up the ramp. How fast, in meters per second, is the block moving right after it comes off the spring?

3.What is the change of the gravitational potential energy, in joules, between the original position of the block at the top of the ramp and the position of the block when the spring is fully compressed?

### Transcription for Educational Website

#### Physics Diagram Explanation: Spring-Mass System on an Inclined Plane

The diagram illustrates a spring-mass system situated on an inclined plane. The key components and their notations are as follows:

1. **Mass (M)**: The mass, denoted by the symbol "M," represents the object placed on the inclined plane. It is depicted in orange.

2. **Inclined Plane**: The plane on which the mass rests is inclined at an angle, denoted by the Greek letter theta (θ).

3. **Spring**: A spring is attached to one end of the inclined plane and is connected to the mass (M).

4. **Distance (d)**: The distance "d" is the length between the equilibrium position of the spring and the current position of the mass.

5. **Length (l)**: The length "l" refers to the length of the spring when it is neither compressed nor stretched.

6. **Inclination Angle (θ)**: The angle of inclination of the plane with respect to the horizontal surface is denoted by theta (θ).

This diagram is useful in studying the concepts of mechanical energy, forces, and motion on an inclined plane, specifically:

- **Potential Energy in the Spring**: Given by the formula \( \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.
- **Gravitational Potential Energy**: The potential energy due to the height of the mass on the inclined plane.
- **Kinetic Energy**: If the mass is in motion, its kinetic energy would also be considered.
- **Forces at Play**: Analysis includes gravitational force (Fg), normal force perpendicular to the plane's surface, and the restoring force exerted by the spring.

This information is critical in understanding the dynamics involved in spring-mass systems on inclined planes, which is a fundamental concept in classical mechanics.

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