/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 Apply How does the potential ene... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 31

Apply How does the potential energy of a spring change if its amount of stretch is doubled?

Short Answer

Expert verified
The potential energy quadruples.

Step by step solution

01

Understanding Potential Energy of a Spring

In a spring, the potential energy is given by the formula \( PE = \frac{1}{2} k x^2 \), where \( PE \) is the potential energy, \( k \) is the spring constant, and \( x \) is the amount of stretch or compression in the spring. This formula shows that the potential energy is proportional to the square of the stretch or compression \( x \).
02

Identifying the Change in Stretch

The problem states that the stretch is doubled. If the original stretch is \( x \), then the new stretch is \( 2x \).
03

Substituting the New Stretch

Substitute \( 2x \) into the original potential energy formula. The new potential energy becomes \( PE' = \frac{1}{2} k (2x)^2 \).
04

Simplifying the New Potential Energy

Calculate \((2x)^2\) which is \(4x^2\). Substitute this into the equation, resulting in \( PE' = \frac{1}{2} k imes 4x^2 = 2kx^2\).
05

Comparing New and Original Potential Energies

The original potential energy was \( rac{1}{2}kx^2 \), and the new potential energy is \( 2kx^2 \). Hence, the new potential energy is 4 times the original potential energy. The potential energy of the spring quadruples when the amount of stretch is doubled.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

spring constant
The spring constant, denoted by the symbol \( k \), is a measure of a spring's stiffness. It is an essential parameter in Hooke's Law, which relates the force exerted by a spring to its displacement.
  • A higher spring constant signifies a stiffer spring that requires more force for the same amount of compression or stretch.
  • Conversely, a lower spring constant indicates a more flexible spring that stretches or compresses more easily under the same force.
This constant has units of force per unit length, typically Newtons per meter (N/m) in the International System of Units. When calculating the potential energy stored in a spring, the spring constant directly influences the outcome, as the energy is proportional to both the constant and the square of the displacement. Understanding this relationship helps us predict how much energy a spring can store when deformed.
compression
Compression in the context of a spring refers to the act of reducing its length by applying a force. This term is used interchangeably with stretch, as both involve changing the spring's original shape to either store or release potential energy.
  • When a spring is compressed, the coils are pushed closer together.
  • Stretching, on the other hand, pulls the coils further apart.
Both compression and stretching involve altering the potential energy of the spring, governed by the formula \( PE = \frac{1}{2} k x^2 \). The variable \( x \) represents the amount of compression or stretch, and it plays a crucial role in energy calculations.Understanding compression is vital in applications like shock absorbers and springs in toys, where controlled energy release is necessary.
quadratic relationship
The quadratic relationship in the spring potential energy formula is a key concept that dictates how energy increases with displacement. The formula \( PE = \frac{1}{2} k x^2 \) shows that the potential energy (PE) is proportional to the square of the change in length \( x \).This means:
  • If you double the displacement of the spring, the resulting energy is not doubled, but is actually quadrupled (since \((2x)^2 = 4x^2\)).
  • Similarly, if the displacement is tripled, the energy increases by a factor of nine.
Understanding this quadratic relationship is crucial for solving problems related to springs, as it explains why even small increases in displacement can lead to significant changes in energy. This principle is widely used in engineering and physics to design systems that efficiently store and release energy.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Apply How does the kinetic energy of an object change if its speed doubles? Triples?

Engine 1 produces twice the power of engine 2 . If it takes engine 1 the time \(T\) to do the work \(W\), how long does it take engine 2 to do the same work? Explain.

Is it possible for the kinetic energy of an object to be negative? Explain.

A \(51-\mathrm{kg}\) packing crate is pulled across a rough floor with a rope that is at an angle of \(43^{\circ}\) above the horizontal. If the tension in the rope is \(120 \mathrm{~N}\), how much work is done on the crate to move it \(18 \mathrm{~m}\) ?

Concept Check Engine 1 does twice the work of engine \(2 .\) Is it correct to conclude that engine 1 produces twice as much power as engine 2? Explain.
See all solutions

Recommended explanations on Physics Textbooks

Physics of Motion

Read Explanation

Magnetism

Read Explanation

Fields in Physics

Read Explanation

Solid State Physics

Read Explanation

Medical Physics

Read Explanation

Radiation

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.