Experiment #6
Hooke's Law and the Work done by a Spring
Purpose:
1- To verify Hooke's law for springs, and determine the spring constant which characterizes the
force exerted by a spring.
2- To verify the relationship the amount of work done by a spring as it is stretched.
Introduction:
If a mass m on a spring is displaced from the equilibrium position (x0 = 0) to a new
position x, Hooke's law states that the spring will exert a restoring force on the mass Fs = -kx.
The "-" sign indicates that the direction of the spring force is in the direction opposite to the
direction of the displacement. The value "k" is a constant for a given spring, but different springs
have different "k-values." Thus, the force exerted by a spring is variable, specifically the greater
distance it is stretched from equilibrium, the greater is the spring force attempting to restore the
spring to its equilibrium position.
Work is a measure of energy input to or energy output from a system. When the force
exerted on a system is constant, the quantity of work done by the force is easy to determine, W =
FgDy, where Dy is the displacement of the system and Fg is the magnitude of the gravitational
force. Work can have a negative value or a positive value. The value is negative when the force
points in the direction opposite to the displacement, and the value of work is positive when the
force points in the same direction as the displacement.
When, as in the case of a spring, the force is variable, the total work done by a force
cannot be simply calculated using this formula. However, the total work can be calculated piece
wise. If you measure the force at many positions, xi, between x0 = 0 and xf, you can estimate the
amount of work done during any interval. The total work done in moving the system from x0 to
xf is then the sum of work done in the little intervals. The estimate of the work done in a small
interval is Wi = (Fave,i)Dxi. For example, the work done by the force in moving the system from
x0 to x1 is W1-0 = [(F1 + F0)/2] (x1 - x0). The work done by the force in moving the system from
x3 to x4 is W4-3 = [[(F4 + F3)/2 ] (x4 - x3). The total work done by the force in moving the system
from 0 to some position can be estimated as Wtotal = ΣWi. As the size of the measured intervals
is made smaller, the accuracy of these estimates get better. For a spring which obeys Hooke's
Law, it can be shown that the total work done in stretching a spring a distance x is equal to W =
0.5kx2.
Procedure:
1. Measure the mass ms of the spring.
2. Determine the range of mass values that you will investigate. The range of masses
should correspond to a wide range of spring extensions. Please don't overstretch the
spring.
3. By hanging mass M on the spring and reading the position of the mass hanger against a
meter stick, you are going to determine the spring constant k. Be sure to use SI units. It is
up to you to decide how many different values of M to use, and by how much to increase
M for each measurement. The mass of the hanger is included as part of M. Note that there
is no y value for M = 0, and that the origin of the meter stick has no particular meaning.
These are not real problems, for reasons given below. However, you should not move the
meter stick after you start making measurements!
4. Make a graph of Fg (vertical axis) vs. y (horizontal axis). [Fg = Mg for the spring when
it is in equilibrium.] Determine the slope of your best line through the data. This slope
should correspond to the spring constant k in N/m. Note that the slope does not depend
on where you placed the origin of the meter stick. Also note that your first few data
points might not be on a straight line. If this is the case, you may ignore these points in
drawing your best line.
5. Take a different type of spring. Stretch it a little to get a feel for the strength of the
spring. Do the same for the first spring you measured. Starting in step 6 you will make
measurements of the force in the second spring. But first, make a prediction of what you
expect to find. What will the graph of Fg vs y look like for the second spring relative to
the first spring? How will the spring constant of the second spring compare to the spring
constant of the first spring? Be sure to identify which spring is which with a clear
description of the two springs.
6. Repeat steps 1 through 4 for the second spring.
7. Compare your results with your predictions. What does the spring constant tell us?
What are the characteristics of springs with large spring constants compared to those
springs with small spring constants?
8. For one spring, calculate the work done by the gravitational force as it stretched the
spring. Do this by first estimating the work done by the force between successive
measurements of the force and then by adding together these individual contributions to
the total work. Compare this value to the predicted by the relationship W = 0.5kx2.