### The spring constant

**Q.** What is the spring constant? How do you calculate it? Does it change if we use a different mass to calculate it?

**A.** The spring constant is a measure of how stiff the spring is. A spring that is very hard to stretch out has a large spring constant. A spring that is easy to stretch has a small spring constant. As the term spring ** constant **implies, the spring constant is always the same for a given spring, assuming you don't put so much force on it that you break it.

In DR we had some springs, a meter stick, and some known masses. You can calculate the spring constant using the equation F = -k*x.

In this equation, the minus sign tells you that if you displace the spring (x) in one direction, the force is in the *opposite *direction. This is because both x and F are vectors and so the minus sign simply reverses the direction.

So, to find k we first need to exert a force on our spring. We do this by placing a mass on the spring. This mass exerts a force F=mg on the spring, its weight. Since after a few seconds we see that the spring and the mass stop moving we know that the net force is 0. The spring must be exerting a force on the mass to hold it up. This force is exactly the same amount of force that the mass is exerting on the spring. Thus, we can conclude that the F in our equation is just the weight that we place on the spring.

For a 500g mass, F = m*g = (0.500 kg)*(10 m/s^2) = 5 N

Notice that I converted the mass from grams to kilograms so that the force would then be in Newtons.

Now we just need to find x. x is the displacement of the spring. If the end of the spring was originally at 30cm and was then at 45cm after the mass was placed on it the displacement is then:

x= 45cm - 30cm = 15 cm = 0.15 m

Again notice I converted, this time form centimeters to meters. It's not absolutely necessary, but it's a good habit to always put things in their standard units.

So now we can calculate k. Since we have F = -kx we can rearrange to get:

k = -F / x

Again, the minus sign is just telling us about the direction. The force the spring is exerting is up while the displacement is down. Plugging in our numbers we have:

k = 5 N / 0.15 m = 33 N/m

If we were to use the same spring and put a 200g mass on it, by how much would it be displaced?

Since we know k is constant, we can find x. F = mg = (0.200kg)(10m/s^2) = 2N

F = -kx

x = - F / k

x = 2N / (33 N/m) = 0.06m = 6cm

As you should have expected this smaller mass stretched the spring less than our original 500g mass.