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)...
Related questions
Question
6.7/6.8. 5
![**Title: Work Done in Stretching and Compressing a Spring on a Horizontal Surface**
### Problem Statement:
A spring on a horizontal surface can be stretched and held 0.4 m from its equilibrium position with a force of 40 N.
1. **Questions:**
a. How much work is done in stretching the spring 2.5 m from its equilibrium position?
b. How much work is done in compressing the spring 1.5 m from its equilibrium position?
2. **Integral Setup:**
Set up the integral that gives the work done in stretching the spring 2.5 m from its equilibrium position. Use increasing limits of integration.
∫[limits] ( ) dx
(Type exact answers.)
### Explanation:
#### Understanding the Problem:
To solve the problem, we need to calculate the work done in stretching and compressing a spring using the concept of work and Hooke's Law. The force needed to stretch or compress a spring is given by Hooke's Law:
\[ F(x) = kx \]
Where:
- \( F \) is the force.
- \( k \) is the spring constant.
- \( x \) is the displacement from the equilibrium position.
#### Step-by-Step Solution:
**Step 1: Determine the Spring Constant (\( k \))**
Given:
\[ x = 0.4 \, m \]
\[ F = 40 \, N \]
Using Hooke’s Law:
\[ F = kx \]
\[ k = \frac{F}{x} = \frac{40 \, N}{0.4 \, m} = 100 \, N/m \]
**Step 2: Calculate the Work Done in Stretching the Spring 2.5 m**
The work done in stretching or compressing a spring is given by:
\[ W = \int_0^x F(x) \, dx \]
\[ F(x) = kx \]
So,
\[ W = \int_0^x kx \, dx = \int_0^{2.5} 100x \, dx \]
Evaluating the integral:
\[ W = 100 \int_0^{2.5} x \, dx = 100 \left[ \frac{x^2}{2} \right]_0^{2.5} = 100 \left( \frac{(2.5)^2}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0bf917ec-3bd5-4ecc-ad53-7da034b3ca56%2Fe84010fe-7d49-4734-b9c5-7315058ca3ff%2Fprwko6.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Work Done in Stretching and Compressing a Spring on a Horizontal Surface**
### Problem Statement:
A spring on a horizontal surface can be stretched and held 0.4 m from its equilibrium position with a force of 40 N.
1. **Questions:**
a. How much work is done in stretching the spring 2.5 m from its equilibrium position?
b. How much work is done in compressing the spring 1.5 m from its equilibrium position?
2. **Integral Setup:**
Set up the integral that gives the work done in stretching the spring 2.5 m from its equilibrium position. Use increasing limits of integration.
∫[limits] ( ) dx
(Type exact answers.)
### Explanation:
#### Understanding the Problem:
To solve the problem, we need to calculate the work done in stretching and compressing a spring using the concept of work and Hooke's Law. The force needed to stretch or compress a spring is given by Hooke's Law:
\[ F(x) = kx \]
Where:
- \( F \) is the force.
- \( k \) is the spring constant.
- \( x \) is the displacement from the equilibrium position.
#### Step-by-Step Solution:
**Step 1: Determine the Spring Constant (\( k \))**
Given:
\[ x = 0.4 \, m \]
\[ F = 40 \, N \]
Using Hooke’s Law:
\[ F = kx \]
\[ k = \frac{F}{x} = \frac{40 \, N}{0.4 \, m} = 100 \, N/m \]
**Step 2: Calculate the Work Done in Stretching the Spring 2.5 m**
The work done in stretching or compressing a spring is given by:
\[ W = \int_0^x F(x) \, dx \]
\[ F(x) = kx \]
So,
\[ W = \int_0^x kx \, dx = \int_0^{2.5} 100x \, dx \]
Evaluating the integral:
\[ W = 100 \int_0^{2.5} x \, dx = 100 \left[ \frac{x^2}{2} \right]_0^{2.5} = 100 \left( \frac{(2.5)^2}{
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 4 images
Knowledge Booster
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.Recommended textbooks for you
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
Physics for Scientists and Engineers
Physics
ISBN:
9781337553278
Author:
Raymond A. Serway, John W. Jewett
Publisher:
Cengage Learning
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…
Physics
ISBN:
9780134609034
Author:
Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:
PEARSON