2. A 4.0 kg block is moving at v₁ = 8 m/s along a frictionless, horizontal surface towards a spring with force constant k = 400 N/m that is attached to a wall. See Figure. The spring has negligible mass. (a) Find the maximum distance the spring will be compressed. (b) If the spring is to compress by no more than 0.15 m, what should be the maximum value of vo?

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)...
icon
Related questions
icon
Concept explainers
Topic Video
Question
100%
**Problem 2:** 

A 4.0 kg block is moving at \( v_0 = 8 \text{ m/s} \) along a frictionless, horizontal surface towards a spring with force constant \( k = 400 \text{ N/m} \) that is attached to a wall. See Figure. The spring has negligible mass. 

(a) Find the maximum distance the spring will be compressed.

(b) If the spring is to compress by no more than 0.15 m, what should be the maximum value of \( v_0 \)?

### Explanation:
This problem involves the concept of energy conservation where the initial kinetic energy of the block is converted into the potential energy of the compressed spring. 

1. **Kinetic Energy**:
   Initial kinetic energy of the block is given by:
   \[
   KE_{\text{initial}} = \frac{1}{2} m v_0^2
   \]
   where \( m = 4.0 \text{ kg} \) and \( v_0 = 8 \text{ m/s} \).

2. **Potential Energy of Spring**:
   The potential energy stored in a compressed spring is given by:
   \[
   PE_{\text{spring}} = \frac{1}{2} k x^2
   \]
   where \( k = 400 \text{ N/m} \) and \( x \) is the compression of the spring.

3. **Maximum Compression**:
   Setting the initial kinetic energy equal to the potential energy of the spring at maximum compression:
   \[
   \frac{1}{2} m v_0^2 = \frac{1}{2} k x^2
   \]
   
4. **Solving Part (a)**:
   Isolate \( x \) to find the maximum compression:
   \[
   x = \sqrt{\frac{m v_0^2}{k}}
   \]
   
5. **Solving Part (b)**:
   Given the maximum compression \( x = 0.15 \text{ m} \), solve for \( v_0 \) by rearranging the energy conservation equation:
   \[
   v_0 = \sqrt{\frac{k x^2}{m}}
   \]

(Note: Ensure according to specific instructions in your educational material to add numerical solutions and final answers as per calculation if required.)
Transcribed Image Text:**Problem 2:** A 4.0 kg block is moving at \( v_0 = 8 \text{ m/s} \) along a frictionless, horizontal surface towards a spring with force constant \( k = 400 \text{ N/m} \) that is attached to a wall. See Figure. The spring has negligible mass. (a) Find the maximum distance the spring will be compressed. (b) If the spring is to compress by no more than 0.15 m, what should be the maximum value of \( v_0 \)? ### Explanation: This problem involves the concept of energy conservation where the initial kinetic energy of the block is converted into the potential energy of the compressed spring. 1. **Kinetic Energy**: Initial kinetic energy of the block is given by: \[ KE_{\text{initial}} = \frac{1}{2} m v_0^2 \] where \( m = 4.0 \text{ kg} \) and \( v_0 = 8 \text{ m/s} \). 2. **Potential Energy of Spring**: The potential energy stored in a compressed spring is given by: \[ PE_{\text{spring}} = \frac{1}{2} k x^2 \] where \( k = 400 \text{ N/m} \) and \( x \) is the compression of the spring. 3. **Maximum Compression**: Setting the initial kinetic energy equal to the potential energy of the spring at maximum compression: \[ \frac{1}{2} m v_0^2 = \frac{1}{2} k x^2 \] 4. **Solving Part (a)**: Isolate \( x \) to find the maximum compression: \[ x = \sqrt{\frac{m v_0^2}{k}} \] 5. **Solving Part (b)**: Given the maximum compression \( x = 0.15 \text{ m} \), solve for \( v_0 \) by rearranging the energy conservation equation: \[ v_0 = \sqrt{\frac{k x^2}{m}} \] (Note: Ensure according to specific instructions in your educational material to add numerical solutions and final answers as per calculation if required.)
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Knowledge Booster
Simple Harmonic Motion
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
College Physics
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
Physics for Scientists and Engineers
Physics for Scientists and Engineers
Physics
ISBN:
9781337553278
Author:
Raymond A. Serway, John W. Jewett
Publisher:
Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:
9780321820464
Author:
Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:
Addison-Wesley
College Physics: A Strategic Approach (4th Editio…
College Physics: A Strategic Approach (4th Editio…
Physics
ISBN:
9780134609034
Author:
Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:
PEARSON