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Chapter 8: Problem 14

Pushing on the pump of a soap dispenser compresses a a small spring. When the spring is compressed \(0.50 \mathrm{cm},\) its potential energy is 0.0025 J. (a) What is the force constant of the spring? (b) What compression is required for the spring potential energy to equal \(0.0084 \mathrm{J} ?\)

Short Answer

Expert verified
(a) The force constant is 200 N/m. (b) The required compression is 0.92 cm.

Step by step solution

01

Understanding Potential Energy of a Spring

The potential energy stored in a compressed or stretched spring is given by the formula: \( U = \frac{1}{2} k x^2 \), where \( U \) is the potential energy, \( k \) is the force constant (also known as the spring constant), and \( x \) is the compression or extension of the spring.
02

Solving for Force Constant (Part a)

Given that the spring potential energy is \( U = 0.0025 \) J when the spring is compressed by \( x = 0.50 \) cm (or 0.005 m), the force constant \( k \) can be found using: \( 0.0025 = \frac{1}{2} k (0.005)^2 \). Solving for \( k \), we get: \( k = \frac{2 \times 0.0025}{(0.005)^2} \approx 200 \) N/m.
03

Determining Required Compression (Part b)

To find the compression \( x \) required for the potential energy to equal \( 0.0084 \) J, use the formula with known \( k = 200 \) N/m: \( 0.0084 = \frac{1}{2} \times 200 \times x^2 \). Solving for \( x \), we get: \( x^2 = \frac{0.0084 \times 2}{200} = 0.000084 \). Therefore, \( x \approx \sqrt{0.000084} \approx 0.009165 \) m or 0.92 cm.

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Key Concepts

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

Spring Constant
The concept of a spring constant, often denoted as \( k \), is fundamental in understanding the behavior of springs in physics. It is a measure of the stiffness of a spring. The spring constant quantifies the amount of force required to compress or extend the spring by a unit distance. A higher spring constant indicates a stiffer spring that requires more force to compress or stretch.
For example, in the given problem, we find the spring constant by using the potential energy formula for springs. The formula is \( U = \frac{1}{2} k x^2 \), where \( U \) is the potential energy, \( k \) is the spring constant, and \( x \) is the displacement. By rearranging this formula, we can solve for \( k \) and determine the stiffness of the spring.
Potential Energy Formula
The potential energy formula for a spring is used to calculate the energy stored in a spring when it is either compressed or stretched. This energy can be found using the equation:\( U = \frac{1}{2} k x^2 \).
Let's break it down:
  • \( U \) represents the potential energy stored in the spring, measured in Joules (J).
  • \( k \) is the spring constant, measured in newtons per meter (N/m).
  • \( x \) is the displacement, or the distance the spring is compressed or extended, measured in meters (m).
This formula shows that the potential energy is directly proportional to the square of the displacement. This means that even a small change in the displacement can result in a significant change in the potential energy of the spring.
Force Constant Calculation
Calculating the force constant is an essential step in analyzing spring mechanisms. The force constant, or spring constant, \( k \) is calculated using the potential energy formula \( U = \frac{1}{2} k x^2 \). By isolating \( k \) in this equation, we can find the spring constant depending on known values of potential energy and displacement.
In our problem, we know the potential energy \( U = 0.0025 \) J and the displacement \( x = 0.005 \) m. Plugging these into the equation, we first rewrite it as \( k = \frac{2U}{x^2} \). After calculating, we find that \( k \approx 200 \) N/m.
This method of calculation can be used for any spring as long as we know two of the three variables: \( k \), \( x \), or \( U \). It provides a clear way to quantify and understand the forces involved in spring compression and extension.

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Most popular questions from this chapter

A 0.21 apple falls from a tree to the ground, \(4.0 \mathrm{m}\) below. Ignoring air resistance, determine the apple's kinetic energy, \(K\) the gravitational potential energy of the system, \(U\), and the total mechanical energy of the system, \(E\), when the apple's height above the ground is (a) \(4.0 \mathrm{m},\) (b) \(3.0 \mathrm{m},\) (c) \(2.0 \mathrm{m}\), (d) \(1.0 \mathrm{m},\) and (e) \(0 \mathrm{m}\). Take ground level to be \(y=0\). (e) \(0 \mathrm{m}\). Take ground level to be \(y=0\).

BIO Compressing the Ground A running track at Harvard University uses a surface with a force constant of \(2.5 \times 10^{5} \mathrm{N} / \mathrm{m}\) This surface is compressed slightly every time a runner's foot lands on it. The force exerted by the foot, according to the Saucony shoe company, has a magnitude of \(2700 \mathrm{N}\) for a typical runner. Treating the track's surface as an ideal spring, find (a) the amount of compression caused by a foot hitting the track and (b) the energy stored briefly in the track every time a foot lands.

Taking a leap of faith, a bungee jumper steps off a platform and falls until the cord brings her to rest. Suppose you analyze this system by choosing \(y=0\) at the platform level, and your friend chooses \(y=0\) at ground level. (a) Is the jumper's initial potential energy in your calculation greater than, less than, or equal to the same quantity in your friend's calculation? Explain. (b) Is the change in the jumper's potential energy in your calculation greater than, less than, or equal to the same quantity in your friend's calculation? Explain.

A sled slides without friction down a small, ice-covered hill. If the sled starts from rest at the top of the hill, its speed at the bottom is \(7.50 \mathrm{m} / \mathrm{s}\). (a) On a second run, the sled starts with a speed of \(1.50 \mathrm{m} / \mathrm{s}\) at the top. When it reaches the bottom of the hill, is its speed \(9.00 \mathrm{m} / \mathrm{s},\) more than \(9.00 \mathrm{m} / \mathrm{s},\) or less than \(9.00 \mathrm{m} / \mathrm{s}\) ? Explain. (b) Find the speed of the sled at the bottom of the hill after the second run.

Running Shoes The soles of a popular make of running shoe have a force constant of \(2.0 \times 10^{5} \mathrm{N} / \mathrm{m} .\) Treat the soles as ideal springs for the following questions. (a) If a \(62-\mathrm{kg}\) person stands in a pair of these shoes, with her weight distributed equally on both feet, how much does she compress the soles? (b) How much energy is stored in the soles of her shoes when she's standing?
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