Question

What two factors are causing the force that stretches the spring? What would happen to the spring if the force were increased? Is the relationship between force and the elastic energy (Eel) a direct or inverse relationship?

Answer 1

1) The spring constant (k) and the displacement (x)

2) The displacement of the spring will increase

3) Direct relationship

Explanation:

1)

The restoring force in a spring is given by Hooke's law:

$F=-kx$

where

F is the force

k is the spring constant

x is the stretching/compression of the spring with respect to its natural length

The negative sign means that the direction of the restoring force is opposite to the direction of the stretching/compression.

From the equation, we see that the two factors affecting the force in the spring are:

- The spring constant, k

- The displacement of the spring, x

2)

We can solve this part by re-arranging the previous equation as

$x=-\frac{F}{k}$

So we see that:

- The displacement of the spring is directly proportional the force applied

- The displacement of the spring is inversely proportional to the spring constant

In this problem, the force applied to the spring has been increased: therefore, since x is proportional to F, we can conclude that the displacement of the spring (stretching or compression) will increase as well.

3)

The elastic potential energy stored in a spring is given by

$U=\frac{1}{2}kx^2$

We can rewrite the displacement x using the equations above

$x=-\frac{F}{k}$

So, the elastic potential energy becomes

$U=\frac{1}{2}k(\frac{F}{k})^2=\frac{1}{2}\frac{F^2}{k}$

Therefore, we see that the elastic potential energy is directly proportional to the square of the force applied. Therefore, the relationship between the two quantities is a direct relationship.

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