Question

A spring is stretched \(5.00 \mathrm{~cm}\) from its equilibrium position. If this stretching requires 30.0 J of work, what is the spring constant?

Short answer

Answer: The spring constant for the given spring is 24000 N/m.

Step-by-step solution

Write down the given information

We are given that the spring is stretched 5.00 cm, which is equivalent to 0.0500 m and the work required to stretch the spring is 30.0 J.

Write down the work-energy theorem for a spring

The work-energy theorem states that the work done is equal to the change in potential energy of the system. In the case of a spring, the potential energy is given by the expression: $$ W = \frac{1}{2} kx^2 $$ where \(W\) is the work done, \(k\) is the spring constant, and \(x\) is the displacement from the equilibrium position.

Plug in the given values and solve for the spring constant

We know the work done (\(W\)) and the displacement (\(x\)). We can plug these values into the work-energy theorem and solve for the spring constant (\(k\)): $$ 30.0 \mathrm{J} = \frac{1}{2} k (0.0500 \mathrm{m})^2 $$ Now we need to solve for the spring constant, \(k\).

Rearrange the equation and solve for the spring constant

To find the value of \(k\), let's first multiply both sides by 2, then divide by \((0.0500 \mathrm{m})^2\): $$ k = \frac{2 \times 30.0 \mathrm{J}}{(0.0500 \mathrm{m})^2} $$ Calculate the value of \(k\): $$ k = \frac{60.0 \mathrm{J}}{0.0025 \mathrm{m^2}} $$ $$ k = 24000 \mathrm{N/m} $$

Write the final answer

The spring constant for the given spring is 24000 N/m.

Key concepts

Understanding the Work-Energy Theorem

The Work-Energy Theorem is a fundamental principle in physics that relates the work done on an object to the change in its kinetic energy. When we apply a force to stretch or compress a spring, we're doing work on the spring. According to the theorem, the work done by the forces acting on an object is equal to the change in kinetic energy of that object.

To put it into context with springs, when you stretch a spring and then release it, the energy you used to pull the spring gets converted into kinetic energy, causing the spring to move back to its original position. However, if the spring doesn't move (as in the case of our exercise), the energy is stored as potential energy, known as elastic potential energy, in the spring. That's what's happening in the exercise provided; the work done to stretch the spring is stored as potential energy.

Potential Energy in Springs

Potential energy is the energy stored within a physical system as a result of the position of the objects within that system. For springs, this is known as elastic potential energy. This kind of energy is directly related to the displacement of the spring from its equilibrium position, that is, how much it has been stretched or compressed.

The potential energy (\(U\text{elastic}\)) stored in a spring is given by \( U_{\text{elastic}} = \frac{1}{2} k x^2 \), where \(k\) is the spring constant and \(x\) is the displacement from equilibrium. This formula comes into play when we want to calculate the spring constant from a given amount of work, as seen in the original exercise. By understanding this concept, students can better understand why the work done to stretch the spring is equivalent to the potential energy stored in it.

Hooke's Law and Spring Constant

Hooke's Law is a principle of physics that states the force needed to extend or compress a spring by some distance scales linearly with respect to that distance. This can be represented by the formula \( F = -kx \), where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the displacement from the spring's original length (equilibrium position).

The negative sign indicates that the force exerted by the spring is in the opposite direction to the force causing the displacement. The spring constant, \( k \), represents the stiffness of the spring—the higher the value of \( k \) the stiffer the spring. Using Hooke's Law, one can understand how the constant plays a crucial role in the force-displacement relationship and provides insights into the physical properties of the spring itself. By calculating the spring constant, we can predict how a spring will respond to varying forces.