### Physics Problem: Collision and Spring Compression#### Given:- A ball of clay with mass \( \frac{M}{8} \).A larger mass, known as \( M \), is attached to a spring with spring constant \( k \). A ball of clay with mass \( \frac{M}{8} \) moves with speed \( v \) towards the larger mass \( M \). Upon collision, the ball of clay sticks to the larger mass, which was initially at rest. ##### Questions:a) What would be the speed of the combination after the collision?b) How much did the spring compress?c) How long did it take for the spring to be compressed due to the collision?#### Diagram Explanation:The diagram below provides a visual representation of the problem setup:- There is a ball of clay labeled \( \frac{M}{8} \) on the left, moving to the right with velocity \( v \).- In the middle, there is a larger mass \( M \) initially at rest.- The mass \( M \) is connected to a spring on the right side, which is fixed to a wall.- The spring has a spring constant \( k \).![Diagram](https://example.com/diagram-url)The diagram represents the initial positions and velocities before the collision.#### Analysis:To solve the given problems, we shall use the principles of conservation of momentum, spring compression, and kinematics.##### a) Speed of the CombinationWe'll use the conservation of momentum to find the speed of the combined mass after the collision.##### b) Spring CompressionWe'll use the conservation of energy to determine how much the spring compresses.##### c) Time for Spring CompressionTo find the time taken for the spring to compress, we shall analyze the simple harmonic motion caused by the spring.This problem touches on key concepts in mechanics, including collisions, momentum, and harmonic motion, making it an excellent example for educational purposes.

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Description: ### Physics Problem: Collision and Spring Compression#### Given:- A ball of clay with mass \( \frac{M}{8} \).A larger mass, known as \( M \), is attached to a spring with spring constant \( k \). A ball of clay with mass \( \frac{M}{8} \) moves with

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Answered: a) What would be the speed of the…

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a) What would be the speed of the combination after the collision? b) How much did the spring compressed c) How long did it take for the spring to be compressed due to the collision?

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ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

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Concept explainers

Simple harmonic motion

Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.

Simple Pendulum

A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.

Oscillation

In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.

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Question

### Physics Problem: Collision and Spring Compression

#### Given:
- A ball of clay with mass \( \frac{M}{8} \).

A larger mass, known as \( M \), is attached to a spring with spring constant \( k \). A ball of clay with mass \( \frac{M}{8} \) moves with speed \( v \) towards the larger mass \( M \). Upon collision, the ball of clay sticks to the larger mass, which was initially at rest.

##### Questions:
a) What would be the speed of the combination after the collision?

b) How much did the spring compress?

c) How long did it take for the spring to be compressed due to the collision?

#### Diagram Explanation:
The diagram below provides a visual representation of the problem setup:

- There is a ball of clay labeled \( \frac{M}{8} \) on the left, moving to the right with velocity \( v \).
- In the middle, there is a larger mass \( M \) initially at rest.
- The mass \( M \) is connected to a spring on the right side, which is fixed to a wall.
- The spring has a spring constant \( k \).

![Diagram](https://example.com/diagram-url)

The diagram represents the initial positions and velocities before the collision.

#### Analysis:
To solve the given problems, we shall use the principles of conservation of momentum, spring compression, and kinematics.

##### a) Speed of the Combination
We'll use the conservation of momentum to find the speed of the combined mass after the collision.

##### b) Spring Compression
We'll use the conservation of energy to determine how much the spring compresses.

##### c) Time for Spring Compression
To find the time taken for the spring to compress, we shall analyze the simple harmonic motion caused by the spring.

This problem touches on key concepts in mechanics, including collisions, momentum, and harmonic motion, making it an excellent example for educational purposes.

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