Physics 1405 Name(s)______
HOOKE’S LAW and SIMPLE HARMONIC MOTION
INTRODUCTION
Any motion that repeats itself in equal intervals of time is called periodic motion. A special form of periodic motion is called Simple Harmonic Motion (SHM). Simple Harmonic Motion is defined as oscillatory motion in which the resultant force on the oscillating body at any instant is directly proportional to its displacement from the rest position and opposite in direction to its motion.
For a spring system, this can be written as F = -kx where F is the resultant force on the object attached to the spring, x is the displacement of the object from equilibrium and k is a constant called the spring constant. The force is a restoring force because it tends to restore the object back to its original position. This relationship is called Hooke’s Law.
If a mass is attached to a spring and then displaced from its rest position and released, it will oscillate around that rest position in simple harmonic motion. The period T of the oscillating system does not depend on the displacement from rest as long as the spring is not overstretched. The period is the time it takes for as system to go through one full oscillation and return to its starting position.
In this lab we will study Hooke’s Law for a mass connected to a spring and then investigate the SHM of the mass on the spring. We will find the spring constant in each case and compare the results.
Part I. HOOKE'S LAW
PROCEDURE
1. Mount the spring so that it hangs vertically. Attach a mass hanger or a small mass and allow the system to stretch to an equilibrium state (the x = L situation in the figure).
2. Place the bottom of the mass hanger or small mass even with a reference point on a meter stick as shown in the figure. This will be your zero for measurements.
3. Now, add masses in units of about 50 g up to a total of about 250 g. In each case, record the mass in kg by dividing grams by 1000. Then calculate the weight (weight = mass times g) in Newtons and record this amount. Record the amount of stretch from your reference point in centimeters for each mass. Divide by 100 to convert the stretch length to meters and record in the table.
Mass in kg / Weight in Newtons(y-axis) / Stretch in cm / Stretch in meters
(x-axis)
0 / 0 / 0
ANALYSIS
Use Excel to plot a graph of Weight vs. Stretch for your spring. Weight should be in Newtons and should be plotted on the y-axis. Plot stretch in meters on the x-axis. Use Excel to find the best fit line. The slope of this line represents the spring constant, k. What is that slope?
Slope = Spring Constant, k ______N/m
Part II. SIMPLE HARMONIC MOTION
One example of simple harmonic motion is the oscillation of a mass on a spring. The period of oscillation depends on the spring constant of the spring and the mass that is oscillating. The equation for the period, T, where m is the suspended mass, and k is the spring constant is given as
We will use this relationship to find the spring constant of the spring and compare it to the spring constant found using Hooke’s Law.
PROCEDURE
1. Use the same vertical spring set-up that was used for Hooke’s Law. Attach a small mass of about 50 g to the spring.
2. Start the spring oscillating by pulling the mass down by a few centimeters and releasing it. Make sure the motion is smooth. Time 10 full oscillations with a stopwatch and record the time in the table. Divide this time by 10 to find the period of oscillation for the mass on the spring.
3. Repeat this process by adding mass in about 20-gram increments and measuring the resulting period for five additional masses. Record the data in the table.
4. Calculate the period squared for each mass and record it in the table.
Hanging mass(kg) / T for 10 oscillations (s) / Period T
(s) / T2
(s2)
ANALYSIS
Use Excel to plot T2 vs. hanging mass. T2 should be in seconds2 (y-axis) and hanging mass should be in kg (x-axis). Use Excel to find the best fit line. The slope of the best fit line will allow you to determine the spring constant, k.
Remember that . If we square both sides of this equation we get:
Since we have plotted T2 vs. m, then the slope of the best fit line is related to the spring constant. The spring constant k = (4p2 /slope)
From your best fit line, determine the slope and the spring constant.
Slope:______m/N
k = (4p2 /slope) = ______N/m
CONCLUSION
Calculate the percent difference between spring constants found in parts I and II. Do this by taking the absolute value of the difference of the two spring constants and dividing by the average of the two spring constants.
% Difference = : ______
Which determination of the spring constant do you think is more accurate? Why?
What are some sources of experimental error for each part?
For at least one of the sources of error that you mentioned, discuss what effect would the error have on your results.
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Cavalli 2008