# Hookes Law and Simple Harmonic Motion

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).

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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.

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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 |Stretch in cm |Stretch in meters | | |(y-axis) | |(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...

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