Question

A spring with spring constant \(k\) is initially compressed a distance \(x_{0}\) from its equilibrium length. After returning to its equilibrium position, the spring is then stretched a distance \(x_{0}\) from that position. What is the ratio of the work that needs to be done on the spring in the stretching to the work done in the compressing?

Short answer

Answer: The ratio of the work done on the spring during stretching to the work done during compression is 1:1.

Step-by-step solution

Write the Hooke's Law

Hooke's Law states that the force required to compress or extend a spring is proportional to the distance it's stretched or compressed from its equilibrium position. Mathematically, this is represented as: F = -k(x-x0) where F is the force applied, x is the current position, x0 is the equilibrium position, and k is the spring constant.

Calculate the work done during compression

The work done on the spring during compression is given by the integral of the force with respect to distance x, from equilibrium position x0 to x0 + x. Mathematically, this can be expressed as: W_compression = integral[F dx] from x0 to x0+x W_compression = integral[-k(x-x0) dx] from x0 to x0+x Now let's evaluate the integral: W_compression = -k [x^2/2 - x0x] evaluated from x0 to x0+x W_compression = -k [(x0+x)^2/2 - x0(x0+x) - x0^2/2 + x0^2] W_compression = -k [x0x + x^2/2]

Calculate the work done during stretching

Similarly, the work done on the spring during stretching can be calculated by the integral of the force with respect to distance x, from x0 to x0 - x. Mathematically, this is expressed as: W_stretching = integral[F dx] from x0 to x0-x W_stretching = integral[-k(x-x0) dx] from x0 to x0-x Now let's evaluate the integral: W_stretching = -k [x^2/2 - x0x] evaluated from x0 to x0-x W_stretching = -k [(x0-x)^2/2 - x0(x0-x) - x0^2/2 + x0^2] W_stretching = -k [x0x + x^2/2]

Calculate the ratio of the work done during stretching to the work done during compression

Now we can find the ratio of the work done during stretching to the work done during compression by dividing W_stretching by W_compression: Ratio = W_stretching / W_compression Ratio = (-k [x0x + x^2/2]) / (-k [x0x + x^2/2]) As we can see, the ratio is equal to 1. Therefore, the ratio of the work done on the spring during stretching to the work done during compression is 1:1.

Key concepts

Spring Constant

The spring constant, denoted as k, is a measure of a spring's stiffness. It is a fundamental parameter in Hooke's Law, which states that the force F needed to extend or compress a spring by some distance x is proportional to that distance.

Hooke's Law can be written as: \[ F = -k(x - x_{0}) \].
Here, x_{0} represents the spring's equilibrium length, and x is the displacement from that equilibrium position. The negative sign indicates that the force is restorative; it acts in the opposite direction of displacement.
When dealing with exercises involving spring constant, understand that a higher spring constant means a stiffer spring, which takes more force to compress or extend. Conversely, a lower spring constant means the spring is easier to stretch or compress.

Work Done on a Spring

Work done on a spring involves the force applied to compress or stretch the spring and the distance over which the force is applied. The equation derived from Hooke's Law for the work done, W, is:\[ W = \frac{1}{2}kx^2 \].
This formula applies to both the compression and stretching of a spring.

Using calculus, one can integrate the force over the distance the spring is stretched or compressed to find the work done. The work done is essentially the area under the force-distance curve. It is key to remember work can be both done to the spring, such as when compressing it, and by the spring, such as when it is released and pushes back to its equilibrium position. The exercise and solution shown exemplify how the integral is evaluated for both compression and stretching and shows that in this particular case, where distance of displacement is equal for both actions, the work done is identical.

Compression and Stretching of Spring

The terms 'compression' and 'stretching' refer to the forces that reduce or increase the length of a spring from its natural length. In compression, the spring is pressed together, while in stretching, it is pulled apart.

These actions on a spring are related to work done and can be described through Hooke's Law. When you compress a spring to a distance x, you perform work on it, storing potential energy in the spring. When you stretch a spring by the same distance, you again perform work and store potential energy. Interestingly, assuming the spring is ideal (meaning it follows Hooke's Law perfectly and has no energy loss mechanisms like friction), the total amount of work done in compressing or stretching the spring is the same, given the distance is constant.
This concept is crucial in understanding how springs behave in mechanical systems and in confirming that energy conservation holds true in such ideal scenarios, as illustrated in the textbook exercise provided.