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Chapter 8: Problem 1

What is the spring constant of a spring that stores \(25 \mathrm{~J}\) of elastic potential energy when compressed by \(7.5 \mathrm{~cm} ?\)

Short Answer

Expert verified
The spring constant is approximately 8889 N/m.

Step by step solution

01

Understand the Problem

We are given that the spring stores 25 J of elastic potential energy when it's compressed by 7.5 cm. We need to find the spring constant, often represented by the letter \( k \).
02

Recall the Energy Formula

The formula for elastic potential energy stored in a spring is \[ U = \frac{1}{2} k x^2 \] where \( U \) is the elastic potential energy, \( k \) is the spring constant, and \( x \) is the compression (or extension) of the spring.
03

Substitute Known Values

Substitute the known values into the energy formula: \[ 25 = \frac{1}{2} k (0.075)^2 \] Note that we convert the compression distance from centimeters to meters, as the standard unit of length in physics is meters.
04

Solve for the Spring Constant

First, calculate the square of the compression: \[ 0.075^2 = 0.005625 \]Now substitute back into the equation: \[ 25 = \frac{1}{2} k \, \times \, 0.005625 \] Multiply both sides by 2 to eliminate the fraction:\[ 50 = k \, \times \, 0.005625 \] Divide both sides by 0.005625 to solve for \( k \):\[ k = \frac{50}{0.005625} \]
05

Calculate the Spring Constant

Perform the division to find \( k \):\[ k = 8888.89 \] Therefore, the spring constant is approximately 8889 N/m.

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Key Concepts

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

Understanding the Spring Constant
The spring constant, symbolized by \( k \), is a unique property of each spring that dictates its stiffness. Imagine a spring that requires a lot of force to be compressed just a little鈥攊t has a high spring constant. Conversely, a spring that easily compresses without much force has a low spring constant.
This concept is crucial because it tells us how much force a spring will exert when it's either compressed or stretched. Specifically, the higher the spring constant, the stiffer the spring.
The spring constant can be determined using the formula for elastic potential energy, provided we know other variables like the energy stored and the amount of compression or extension.
The Role of Compression in Spring Mechanics
Compression in terms of springs refers to the act of pressing the spring together, reducing its length. It is a critical variable in determining the elastic potential energy stored within a spring.
When dealing with problems involving elastic potential energy, like the given exercise, compression directly affects the outcome.
  • Compression impacts how much potential energy is stored鈥攖he more the spring is compressed, the more energy it holds.
  • It is usually measured in meters. In physics, it's important to convert units, such as changing centimeters to meters, for consistency.
Remember, compression is not just about physical shortening; it plays a significant part in calculations involving the spring constant and energy storage.
How the Energy Formula Relates to Springs
The energy formula is pivotal in calculating the spring's potential energy. Elastic potential energy is the energy stored when a spring is either compressed or stretched, articulated through the equation:\[ U = \frac{1}{2} k x^2 \]Here:
  • \( U \) represents the elastic potential energy.
  • \( k \) stands for the spring constant, indicating the spring's stiffness.
  • \( x \) denotes the compression or extension of the spring in meters.
By using this formula, one can determine how much energy is stored in a spring for a given amount of compression or extension.
This formula is a foundational concept in fields involving mechanics and physics, allowing the transfer and calculation of energy based on how a spring behaves under force.

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Most popular questions from this chapter

When a click beetle is upside down on its back, it jumps upward by suddenly arching its back, transferring energy stored in a muscle to mechanical energy. This launching mechanism produces an audible click, giving the beetle its name. Videotape of a certain click-beetle jump shows that a beetle of mass \(m=4.0 \times 10^{-6} \mathrm{~kg}\) moved directly upward by \(0.77 \mathrm{~mm}\) during the launch and then to a maximum height of \(h=0.30 \mathrm{~m}\). During the launch, what are the average magnitudes of (a) the external force on the beetle's back from the floor and (b) the acceleration of the beetle in terms of \(g\) ?

A block slides along a path that is without friction until the block reaches the section of length \(L=0.75 \mathrm{~m}\), which begins at height \(h=2.0 \mathrm{~m}\) on a ramp of angle \(\theta=30^{\circ} .\) In that section, the coefficient of kinetic friction is \(0.40 .\) The block passes through point \(A\) with a speed of \(8.0 \mathrm{~m} / \mathrm{s}\). If the block can reach point \(B\) (where the friction ends), what is its speed there, and if it cannot, what is its greatest height above \(A\) ?

A spring with a spring constant of \(3200 \mathrm{~N} / \mathrm{m}\) is initially stretched until the elastic potential energy of the spring is \(1.44 \mathrm{~J}\). \((U=0\) for the relaxed spring.) What is \(\Delta U\) if the initial stretch is changed to (a) a stretch of \(2.0 \mathrm{~cm},(\mathrm{~b})\) a compression of \(2.0 \mathrm{~cm}\), and (c) a compression of \(4.0 \mathrm{~cm}\) ?

The string is \(L=120 \mathrm{~cm}\) long, has a ball attached to one end, and is fixed at its other end. A fixed peg is at point \(P\). Released from rest, the ball swings down until the string catches on the peg; then the ball swings up, around the peg. If the ball is to swing completely around the peg, what value must distance \(d \mathrm{ex}\) ceed? (Hint: The ball must still be moving at the top of its swing. Do you see why?)

Shows an \(8.00 \mathrm{~kg}\) stone at rest on a spring. The spring is compressed \(10.0 \mathrm{~cm}\) by the stone. (a) What is the spring constant? (b) The stone is pushed down an additional \(30.0 \mathrm{~cm}\) and released. What is the elastic potential energy of the compressed spring just before that release? (c) What is the change in the gravitational potential enErgy of the stone-Earth system when the stone moves from the release point to its maximum height? (d) What is that maximum height, measured from the release point?
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