hooke's law and simple harmonic motion (1)

7/27/2019 Hooke's Law and Simple Harmonic Motion (1)

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Physics 1405 Name(s)_____________________

HOOKES 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 asoscillatory motion in which the resultant force on the oscillating body at any instant is directlyproportional 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 attachedto the spring, x is the displacement of the object from equilibrium and kis a constant called the springconstant. The force is a restoring force because it tends to restore the object back to its originalposition. This relationship is called Hookes Law.

If a mass is attached to a spring and then displaced from its rest position and released, it will oscillatearound 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 timeit takes for as system to go through one full oscillation and return to its starting position.

In this lab we will study Hookes Law for a mass connected to a spring and then investigate the SHMof 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 asshown 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 massin kg by dividing grams by 1000. Then calculate the weight (weight = mass times g) in Newtonsand record this amount. Record the amount of stretch from your reference point in centimeters foreach 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 ofWeight vs. Stretch for your spring. Weight should be in Newtons andshould 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

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One example of simple harmonic motion is the oscillation of a mass on a spring. The period ofoscillation depends on the spring constant of the spring and the mass that is oscillating. The equation forthe period, T, where m is the suspended mass, and kis the spring constant is given as

k

m2T

We will use this relationship to find the spring constant of the spring and compare it to the springconstant found using Hookes Law.

PROCEDURE

1. Use the same vertical spring set-up that was used for Hookes Law. Attach a small mass of about50 g to the spring.

2. Start the spring oscillating by pulling the mass down by a few centimeters and releasing it. Make surethe 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 periodfor 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)

Tfor 10 oscillations(s)

Period T(s)

T2

(s2)

ANALYSIS

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Use Excel to plot T2 vs. hanging mass. T2 should be in seconds2 (y-axis) and hanging mass should be inkg (x-axis). Use Excel to find the best fit line. The slope of the best fit line will allow you to determinethe spring constant, k.

Remember thatk

m2T . If we square both sides of this equation we get:

mkT

2

2 4=

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 = (42 /slope)

From your best fit line, determine the slope and the spring constant.

Slope:____________ m/N

k = (42 /slope) = __________N/m

CONCLUSION

Calculate the percent difference between spring constants found in parts I and II. Do this by taking theabsolute value of the difference of the two spring constants and dividing by the average of the twospring constants.

% Difference =%100

)2

( 21

21

+

kk

kk

: ___________

Which determination of the spring constant do you think is more accurate? Why?

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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 onyour results.

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hooke's law and simple harmonic motion (1)

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