Hooke's Law and the Simple Harmonic Motion of a Spring Lab

The purpose of this lab is to find the force constant of a spring and to also study the motion of a spring with a hanging mass when vibrating under the influence of gravity.

INTRODUCTION:
We first need to understand how to calculate the force of a spring before performing this lab. We look at Hooke's Law which can be described as a spring stretched a distance, x, from its equilibrium position will exert a restoring force that is directly proportional to the distance. The force can be written as:

F = -kx

where k is the spring constant and depends on the stiffness of the spring. The minus sign means that the Force is opposite the direction of the displacement of the spring.

To solve for k:

When a mass hangs from a spring, is stretched, and let go, it will oscillate about the equilibrium point of when the mass hangs. From Newton's Second Law and with calculus we can find the period, T (in seconds), from the equation:

where m is the mass supported by the spring

This equation will be altered to solve for k:

Now that we know what our equations we are going to be working with, we need to see where we are going to get these values from:

In this picture we can collect a good amount of information. The x in Hooke's Law can be found in the picture on the left. When the spring is left alone, the spring will not be stretched out at all. When we hang a mass on it, the spring will stretch and whatever the difference is will be the x value (in meters). In the picture on the right, we can see how we will start the second part of the lab which is finding k in an oscillating motion. The mass will be pulled down and oscillate around the equilibrium point of the hanging mass on the spring.

PROCEDURE:
To begin this lab, we will first need to gather the proper materials:

• spring
• masses
• weight hanger
• meter stick
• support stand with clamps
• motion detector
• Lab Pro Interface
• wire basket

We set up our support rods the same way as the picture above and set the motion detector directly underneath the hanging spring inside the wire basket to protect the detector. We hung the spring on the rod and measured how high off the ground the bottom of the spring was.We hung a 350 g mass on the spring and measured how far the bottom of the spring was from the ground with the meter stick. We found the difference in the length of the spring and recorded the value. We then did it for 450 g, 550 g, 650 g, 750 g, 850 g, 950 g, and 1050 g. We then calculated the normal force of the masses which is equal to the weight force since the (Fnet)y was 0. We will use these values to find the spring constant, k, with the equation:

k = F/-x

With these values we were able to make a graph and were able to solve for k doing a linear fit since the k value is the slope.

We plotted the points of the Force applied on the spring vs the change in the length in the spring. We could see that the plot created a linear graph so we performed a linear fit of the points. The slope was -24.5 kg/s^2 and this was our spring constant.

Next, we opened up the file Hooke's Law in Logger Pro because now we are going to stretch the spring with a mass, in equilibrium, 10 cm and allow it to oscillate. We recorded the position of the oscillating mass with the motion detector and allowed it to oscillate for 10 seconds. This gives us the time necessary to perform 5 cycles because in this part of the lab we will be finding an average time, T (sec), per period of the oscillation. We first hung 350 g on the spring and then stretched it 10 cm down and let it go and run for 10 seconds. We then performed this test for all the same masses used in the previous test.

This is the graph of the oscillating mass of 1.050 kg on the spring. From it we performed a sinusoidal fit and could now find the amplitude as well as  the period. The amplitude can be found from the A value given in the Auto Fit of the graph which was 0.07 m or 7 cm. The period of the graph can be found using the equation 2PI/B where B is the value given in the graph. The B value is 1.28 sec/cycle. The value we found for the period from using an average was 1.29 sec/cycle.

With the information obtained above, we were able to make a T^2 vs mass graph.

From this graph we can confirm equation 2 because we can compare it to the equation of a line y=mx+b. The y would be T^2 in this situation because it changes with respect to x, the slope, m, can be found from (4PI^2/k) because it will always be constant. The x value, which is the variable of the function is the mass since the mass is what is changing. The b value should be 0 because this line should be proportional to every single T^2 vs mass all the way to 0.

We can find the value for k from equation 2 since we know the slope of the graph.

CONCLUSIONS:

In this lab I learned how we could find the spring constant of a spring from 1. Hanging a mass and measuring the displacement of the spring and 2. Having a mass oscillate on the spring.

One factor that could have caused a difference in our values for k is that the oscillating mass could have lost some potential energy causing the spring to oscillate slower. We can further study spring constants by studying a spring that compresses and finding its constant.