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# Hooke's Law: Calculating Spring Constants

Forces cause objects to move or deform in some way. **Newton’s third law** states that for every force, there is an equal and opposite force. This is true for springs, which store and use mechanical energy to do work.

Springs are **elastic**, which means after they are deformed (when they are being stressed or compressed), they return to their original shape. Springs are in many objects we use on a daily basis. They in ball point pens, mattresses, trampolines, and absorb shock in our bikes and cars. According to the Third Law of Motion, the harder your pull on a spring, the harder it pulls back. Springs obey **Hooke’s Law**, discovered by Robert Hooke in the 17^{th} century. Hooke’s law is described by:

*F = -kx*

Where *F* is the force exerted on the spring in Newtons (N),

*k *is the spring constant, in Newtons per meter (N/m),

and *x* is the displacement of the spring from its equilibrium position.

The spring constant, *k*, is representative of how stiff the spring is. Stiffer (more difficult to stretch) springs have higher spring constants. The displacement of an object is a distance measurement that describes that change from the normal, or equilibrium, position.

### Problem

Calculate the spring constant using Hooke’s law.

Which spring do you think will have the greatest spring constant? The smallest spring constant? Why?

### Materials

- Scale (measures grams or kilograms)
- Ruler (measuring centimeters)
- Different coil springs
- Small weight
- Wooden plank
- Table or countertop
- Books, or other stackable objects

### Procedure

- With the help of an adult, fix one end of each spring to one side of the wooden plank. Be sure to leave a couple of inches between each spring.
*Why should one end of the spring be fixed?* - Arrange some books on a table or countertop in two stacks, about the length of the wooden plank.
- Place the wooden plank on the stacks with the springs hanging down. Make sure there is still some room between the bottom of the springs and the table.
- Using the centimeter side of a ruler, measure the
**equilibrium position**of each spring. - Weigh the small weight on the scale and record its mass in kilograms.
*Why does the mass have to be in kilograms?* - Attach the weight to each spring one at a time, and use the ruler to measure the
**displacement***.*An easy way to do this is to measure the length of the spring, and then subtract the equilibrium length. - Calculate the gravitational force exerted by the mass on the spring.

*F _{g }= mg*

Where *F _{g }*is the gravitational force, in Newtons, m is the mass of the weight, in kilograms, and g is the gravitational constant of Earth, equal to 9.81 m/s

^{2}.

Set the gravitational force (*F _{g}*) equal to the force exerted by the spring (

*F)*.

*Why can you make these two variables equivalent?*Use Hooke’s law to calculate the spring constant,

*k*, for each spring.

### Results:

Springs with larger spring constants will have smaller displacements than springs with lesser spring constants for the same mass added.

### Why?

Hooke’s Law is a representation of **linear elastic** deformation. Elastic means that the spring will return to its original form once the outside force (the mass) is removed. Linear describes the relationship between the force and the displacement. The fact that the spring constant is a constant (it is a property of the spring itself), shows that the relationship is linear.

Of course, Hooke’s Law only remains true when the material is elastic. If a spring is permanently deformed (by something like crushing or overstretching), it will no longer return to its original position. If you have ever played with a slinky and accidentally stretch it too far or bent it out of shape, you’ll know that it doesn’t perform like it is supposed to afterward.

For Hooke’s Law to work properly, the parts of the equation have to be in the correct units. Without consistent units, the equation is meaningless.

You can set the gravitational force exerted by the mass on the spring equal to the force exerted by the spring due to Newton’s Third Law of Motion, which states that forces come in pairs. Every force has an equal and opposite force.

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