Physics 110: 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 (1)

where F is the resultant force on the object attached to the spring, xis 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. I will add five points to your lab if you highlight this sentence. 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.

In Part II, you will be utilizing Simple Harmonic Motion to determine the spring constant of the spring. 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

(2)

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.

Simulation:

Part I. HOOKE'S LAW

PROCEDURE

1. Navigate to the simulation. Use the "Lab" option in the beginning. In the menu on the right set Damping to "lots" so the springs will not oscillate.
2. Choose a spring constant. You will do this by moving the slider in the middle that goes from small to large.(This allows you to change the stretchiness of the spring) Make sure you note what spring constant you choose, because you will need to get back to the same spring constant for the second part of the lab.
3. Next click on the Displacement/Natural Length box in the menu on the top right. Use the ruler provided in the simulation (located at the bottom right) to measure the unstretched or natural length of the spring. Record the unstretched length below.
4. Now add a mass to your spring. It defaults to 100 grams as your initial mass; however, depending on the spring constant chosen, you may need to change the initial mass depending on the stiffness of the spring. Record in Table 1 the mass and the amount of stretch from the natural length of the spring.
5. Now, change the hanging masses at appropriate intervals. (For a stiffer spring you should use larger increments). In each case, record the mass in kg. Then calculate the force of gravity or weight (F_g = m*g) in and record this amount. Record the amount of stretch from your reference point in meters for each mass.

1. Now change the spring constant of the spring so that you have a noticeably different stiffness in the spring.
2. Repeat steps 1-5 for this new spring constant.

Spring 1

Spring 1 end position (m): ____________________________

Mass (kg)	Weight (N)	Stretch (m)
Table 1: Spring 1 Hooke's Law

Spring 2

Spring 2 end position (m): ____________________________

Mass (kg)	Weight (N)	Stretch (m)
Table 2: Spring 2 Hooke's Law

5. Plot a graph of Stretch vs. Forcefor each of the springs. The slope of this line represents the inverse of the spring constant, 1/k. What is that slope? Show Work (if you plotted on excel just write plotted on excel)!

Slope = _______________ Spring Constant Spring 1 , k _________N/m

Slope= ___________________________ Spring Constant Spring 2 , k _________N/m

Show Your Work:

Part II. SIMPLE HARMONIC MOTION

PROCEDURE B

1. Return to the spring constant chosen initially in Procedure A.
2. Set the damping to none.
3. Add a mass to the spring in order to start it oscillating.
4. Using either the timer provided in the simulation (located next to the ruler on the bottom right) or the stopwatch on your phone, time how long it takes for the spring to complete 10 oscillations. (An oscillation is a complete up and down motion) Record the time in the table.Divide this time by 10 to find the period of oscillation for the mass on the spring. Record the period in the table.
5. 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.
6. Calculate the period squared for each mass and record it in the table.
7. Repeat steps 1-6 with the second spring constant chosen from Procedure A.

Table 3: Procedure B: Spring 1

Hanging mass (kg)	Tfor 10 oscillations (s)	Period T (s)	T2 (s2)

Table 4: Procedure B: Spring 2

Hanging mass (kg)	Tfor 10 oscillations (s)	Period T (s)	T2 (s2)

ANALYSIS

Use Excel to plot T2vs. hanging mass. T2should be in seconds2(y-axis) and hanging mass should be in kg (x-axis). 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 T2vs. 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.

Spring 1

Slope:____________ m/N

k = (4p² /slope) = __________N/m

Spring 2

Slope:____________ m/N

k = (4p² /slope) = __________N/m

1. Calculate the percent difference between spring constants found in Parts 1 and II for spring 1.

(Show your work)

1. Calculate the percent difference between spring constants found in Parts 1 and II for spring 2.

(Show your work)

1. Which determination of the spring constant (Hooke's Law or SHM) do you think is more accurate? Why?

1. What are some sources of experimental error for each part? (Human error or calculations are not sufficient responses).For at least one of the sources of error discuss what effect would the error have on your results.

ANSWER ALL QUESTIONS AND TABLES 1-4

https://phet.colorado.edu/en/simulation/mass-spring-lab