Slide 1

LAB 10: SIMPLE HARMONIC MOTION

Please list the members of your group:
Adri Lila and Zach O'Rourke

OBJECTIVES
To understand two ways to measure the spring constant of a spring.
To understand simple harmonic motion.
To be able to describe the forces involved in the motion of a pendulum.

OVERVIEW
Simple harmonic motion (SHM) is a type of motion of a body that is periodic: it repeats itself. The term
SHM describes various kinds of motions. In a mechanical systems SHM occurs when the force on a
body is (1) proportional to the displacement of the body from its equilibrium position and (2) acts in a
direction opposite to that of the displacement. SHM is a model that describes the oscillation of a mass
hanging from a spring as long as the mass of the spring is negligible compared to the hanging mass.
The motion of a simple pendulum—e.g., a mass swinging at the end of a string—is also approximately
described as SHM under certain circumstances.

The oscillation of a mass hanging from a spring is SHM since it is subject to the linear elastic restoring
force as given by Hooke's law. The motion is sinusoidal in time. In the case of the simple pendulum, the
restoring force on the bob—the object that is swinging—is the gravitational force.

In this lab you will begin by exploring SHM with a spring, and the dependence of the period on the
spring constant of the spring and the mass. Next, you will explore a pendulum system and the
dependence of the period of pendulum on the length of the string and the mass of the bob. You will
explore whether or not mechanical energy is still conserved in such systems.

Copyright © 2018 John Wiley & Sons, Inc.

Slide 2

INVESTIGATION 1: SHM AND ELASTIC POTENTIAL

ENERGY
You have seen in Lab 9 that the magnitude of the force applied by most springs is proportional to the
amount the spring is stretched or compressed beyond its unstretched length. This is usually written:
Fspring = -kx, where k is called the spring constant.

The spring constant can be measured by applying measured forces to the spring and measuring its
extension. In the diagram below, the applied force is shown. By the third law, the force applied is F = kx
as is shown in the diagram.

You also saw in Lab 9 that the work done by a force can be calculated from the area under the force vs.
position graph. shown below is a force vs. position graph for a spring. Note that k is the slope of this
graph, i.e., it is how much the force increases (in newtons) for a 1 meter increase in the amount the
spring is stretched.

Question 1-1: How much work is done in stretching a spring of spring constant k from its unstretched
length by a distance x? (Hint: Look at the triangle on the force vs. position graph above and remember
how you calculated the work done by a changing force in Lab 9.)

The area under the graph

To measure the work done in stretching a spring in the following activity, you will need:

IOLab and software with calibrated force sensor

Spring, unstretched about 7 cm long
Smooth board or other level surface at least 0.5 m long that can be inclined
Meter stick or measuring tape
Slide 3
Force [Fᵧ] vs Wheel - Position [rᵧ] (100 Hz) Swap axis
0.0

-0.5

-1.0
Force (N)

-1.5

-2.0

-2.5

-3.0

-0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22
Wheel - Position (m)

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0


Activity 1-1: Spring Constant

In this Activity, you will use the wheel and the force sensor to measure the spring constant. By moving
the IOLab from the essentially un-stretched (just taut) length of the spring and zero force, you can make
a graph of force vs. position (the amount the spring was stretched).

1. Use the IOLab on a level surface.

2. Measure the force as a function of the position as the IOLab is moved such that the spring
gradually stretches from 0 to 0.2 m . Do not exceed a force larger than 15 N.
3. Use the following two ways to determine the spring constant from your data:
Method 1: Find the spring constant by plotting a graph of force vs. position using the
parametric plot feature in the IOLab software (that you used previously in Lab 9), and
calculating the slope (rise over run).
Method 2: Record the force and spring extension for four different positions of the IOLab
below. Plot a graph in Excel of force vs. position from these four data points. Again, the slope
is the spring constant.
Force (N) Position (m)

-0.5 0

-0.9879 0.0405

-1.6305 0.10
 -2.6981 0.18

Spring constant from parametric plot:

-12.2

Spring constant from Excel:

-12.128

Question 1-2: Was the force exerted by the spring proportional to the displacement of the spring?

Yes No

Question 1-3: If two springs are stretched different amounts by the same mass hung from them, which
spring has the larger spring constant, the one that stretches most or the one that stretches least?
Explain.

The spring that stretches the least will have the greater spring constant.
Slide 4

Activity 1-2: Determine the spring constant again

Prediction 1-1: Suppose the IOLab is hanging from the spring and the force sensor is re-zeroed at the
equilibrium position. As the IOLab oscillates up and down, what would the direction of the force be as
measured by the force sensor (a) when the mass is above the equilibrium position and (b) when it is
below the equilibrium position? (Note that since the mass oscillates up and down around its new
equilibrium position, you don’t need to include the gravitational force.)

a) The force would be downwards

b) The force would be upwards.

Prediction 1-2: As the IOLab oscillates up and down, how will the period change as the amplitude is
changed?

The period will get smaller as the amplitude decreases.

To test your predictions in the following activities, you will need:

IOLab and software

Spring and large binder clip
Plank or board setup almost vertically
Slide 5
Force (200 Hz) Remote 1
∆t: 8.39000 s
μ: 1.738 N — σ: 0.22 N a: 14.582 Ns s: -0.01 N/s (r²: 0.01)
2.5
μ: 1.788 N — σ: 0.37 N a: 15.004 Ns s: -0.01 N/s (r²: 0.01)
Fᵧ (N)

2.0
1.5
1.0
0 5 10 15 20 25 30
  Rezero sensor Time (s)

Wheel - Position (100 Hz) Remote 1

∆t: 8.38559 s
0.15 μ: 0.054 m — σ: 0.021 m a: 0.454 ms s: 0.00 m/s (r²: 0.00)
μ: 0.080 m — σ: 0.036 m a: 0.668 ms s: 0.00 m/s (r²: 0.01)
rᵧ (m)

0.10
0.05
0.00
0 5 10 15 20 25 30
  Rezero sensor Time (s)

Wheel - Velocity (100 Hz) Remote 1

∆t: 8.38559 s
0.6 μ: 0.007 m/s — σ: 0.16 m/s a: 0.056 m s: -0.00 m/s² (r²: 0.00)
0.4
vᵧ (m/s)

0.2 μ: 0.008 m/s — σ: 0.26 m/s a: 0.067 m s: -0.01 m/s² (r²: 0.01)
-0.0
-0.2
-0.4
-0.6
0 5 10 15 20 25 30
  Time (s)

Wheel - Acceleration (100 Hz) Remote 1

3
∆t: 8.38559 s
2 μ: 0.001 m/s² — σ: 1.2 m/s² a: 0.008 m/s s: -0.00 m/s³ (r²: 0.00)
aᵧ (m/s²)

1 μ: 0.002 m/s² — σ: 1.9 m/s² a: 0.018 m/s s: -0.00 m/s³ (r²: 0.00)
0
-1
-2
-3
0 5 10 15 20 25 30
  Time (s)

Activity 1-2: Determine the spring constant again

Attach the spring to the eyebolt and screw the eyebolt into the force sensor. Set up the plank such that
it is practically vertical against a wall or table. (See diagram below.) Affix the spring securely to the top
of the ramp. Unhook one of the arms of the large binder clip and attach the long spring and reattach the
arm. The wheels should be in touch with ramp. You will first need to find the new equilibrium position of
the spring with the IOLab hanging from the spring resting on its wheels against the almost vertical
ramp.
1. Suspend the IOLab from the spring and be sure it is at rest.
2. Start the IOLab oscillating with an amplitude of about 5 cm. Take care that the wheels stay in
touch with the ramp and that the motion of the IOLab is up and down and not sideways.
Record.
3. Determine the period of the oscillations by counting the time for a known number of
oscillations, say 10. Highlight ten oscillations with the analysis tool. The period is then the
total time divided by ten. Explain how you determined the period, and show any calculations.
T1 (seconds):
7.855/10 = 0.786
FInd the total time and divide by # of oscillations.

4. Repeat with a larger amplitude (say 1.5 times as big), and again determine the period.
T2 (seconds):
0.839
Slide 6
Force (200 Hz) Remote 1
∆t: 8.39000 s
μ: 1.738 N — σ: 0.22 N a: 14.582 Ns s: -0.01 N/s (r²: 0.01)
2.5
μ: 1.788 N — σ: 0.37 N a: 15.004 Ns s: -0.01 N/s (r²: 0.01)
Fᵧ (N)

2.0
1.5
1.0
0 5 10 15 20 25 30
  Rezero sensor Time (s)

Wheel - Position (100 Hz) Remote 1

∆t: 8.38559 s
0.15 μ: 0.054 m — σ: 0.021 m a: 0.454 ms s: 0.00 m/s (r²: 0.00)
μ: 0.080 m — σ: 0.036 m a: 0.668 ms s: 0.00 m/s (r²: 0.01)
rᵧ (m)

0.10
0.05
0.00
0 5 10 15 20 25 30
  Rezero sensor Time (s)

Wheel - Velocity (100 Hz) Remote 1

∆t: 8.38559 s
0.6 μ: 0.007 m/s — σ: 0.16 m/s a: 0.056 m s: -0.00 m/s² (r²: 0.00)
0.4
vᵧ (m/s)

0.2 μ: 0.008 m/s — σ: 0.26 m/s a: 0.067 m s: -0.01 m/s² (r²: 0.01)
-0.0
-0.2
-0.4
-0.6
0 5 10 15 20 25 30
  Time (s)

Wheel - Acceleration (100 Hz) Remote 1

3
∆t: 8.38559 s
2 μ: 0.001 m/s² — σ: 1.2 m/s² a: 0.008 m/s s: -0.00 m/s³ (r²: 0.00)
aᵧ (m/s²)

1 μ: 0.002 m/s² — σ: 1.9 m/s² a: 0.018 m/s s: -0.00 m/s³ (r²: 0.00)
0
-1
-2
-3
0 5 10 15 20 25 30
  Time (s)

Activity 1-2: Determine the spring constant again

T = 2π√⎯⎯⎯mk⎯. Using the spring constant you measured previously and the mass of the IOLab you also
Question 1-4: For simple harmonic motion of a mass on a spring, the period can be calculated from

measured previously, calculate the period of oscillations. show your calculation.

T (seconds):
T = 2pisqrt(m/k)
T = 2pisqrt(0.200kg/12.128) = 0.806s
How does this calculated period compare to the experimentally determined one?

They're almost the same

Question 1-5: Does the period appear to depend on the amplitude of the oscillations? Explain.

Yes, the bigger the amplitude of the oscillations the more time it takes to reach equilibrium

Question 1-6: How do the directions of the force and acceleration compare? Can you explain why?

acceleration and force are in the same direction because the net force causes the
acceleration to be in the same direction

Question 1-7: How do the magnitudes of the force and acceleration compare? Can you explain why?

their magnitudes are different because the mass is so small the acceleration has to be
bigger than the force
Slide 7

INVESTIGATION 2: THE ACCELEROMETER

In this investigation you will explore the accelerometer you used briefly in Lab 1. You will use the
accelerometer in Investigation 3 to explore the motion of a simple pendulum—a mass hanging from a
string—which is a SHM under certain conditions.

This accelerometer is an electronic device that measures “proper” acceleration, which is the
acceleration it experiences relative to free fall and is the acceleration felt by people and objects.
Accelerometers are very common: every smartphone has one.

You do well to understand how acceleration sensors work before using them in the lab or in your
smartphone. The accelerometer is, in essence, a hollow square box with a small mass or little ball
attached to six springs; the other ends of the springs are each attached to one of the sides of the box.
The mass can therefore move in all directions. Sensing is done electrically to determine the position of
the mass. Any displacement from the center indicates that there is a force acting on the mass. It is the
net force applied by the walls on the mass that is measured.

The accelerometer measures 0 m/s2 when in free fall, despite the fact that it is accelerating. However, to
the mass inside the accelerometer it looks like it is not accelerating. The mass is like a passenger in a
diving airplane: the passenger will be floating in the air and no forces will be measured by her/him. To
an observer in the plane the passenger is not accelerating, but to a bystander on the ground it is clear
s/he is accelerating.

Prediction 1-1: If you put the IOLab on the table with the z-axis pointing up, what will be the acceleration
measured by the IOLab?

0 m/s^2

Prediction 1-2: If you put the IOLab on the table with the y-axis pointing up, what will be the acceleration
measured by the IOLab?

0m/s^2
Slide 8
Accelerometer (400 Hz) Remote 1 Ax Ay Az

20

15

10

5
a (m/s²)

-5

-10

-15

-20
0 5 10 15 20 25 30
  Time (s)

Activity 2-1: The Accelerometer

The following activities should help you to see whether your predictions make sense.

1. Put the IOLab on the table with the z-axis or the wheels pointing up. What do the x, y, and z
components of the accelerometer register?
ax (m/s²):
0

ay (m/s²):
0

az (m/s²):
10

2. While recording with the accelerometer, turn the IOLab such that the y-axis points up. What
do the x, y, and z components of the accelerometer register?
ax (m/s²):
0

ay (m/s²):
10

az (m/s²):
0

What did you observe while turning the IOLab? Did it matter how fast you turned it or how you turned
it?
 No in the end the side that was pointing up measured 10m/s^2

Note: The accelerometer at rest on the earth will measure an acceleration of one g (g = 9.81m/s2)
upwards. The springs exert a net force on the mass holding it up and this is what is measured by the
accelerometer.
Slide 9

Activity 2-2: An Object in Free fall and Tossed Up into the Air
The center of mass or gravity is located under the red “G” label on the front of the IOLab. It is
approximately in the middle of the IOLab. The accelerometer is located under the red label with the “A”.

Prediction 2-1: What acceleration would you measure when you drop the IOLab and before it hits the
ground? Explain. (Of course, you will not be dropping it on the ground.)

9.8 m/s^2

Prediction 2-2: What acceleration would you measure when you toss the IOLab up in the air so that it
does not spin? Explain.

9.8m/s^2

The following activities should help you to see whether your predictions make sense.
Slide 10
Accelerometer (400 Hz) Remote 1 Ax Ay Az

40
35
30
25
20
a (m/s²)

15
10
5
0
-5
-10
-15
3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6
  Time (s)

Activity 2-2: An Object in Free fall and Tossed Up into the Air
CAUTION: In the next activities, make sure that you catch the IOLab.
1. Drop the IOLab without it spinning.
2. At what time did you release the IOLab?

2.4406s

Question 2-1: How do you know?

That is when the y axis acceleration changed from 9.8 to 0.

3. Now toss the IOLab up in the air without it spinning and measure the components of the
acceleration.
4. Indicate the instant that you started the toss and the moment you released the IOLab.

Starting time of your toss:

3.711s

Time of release:
3.90s

Question 2-2: How can you tell?

 The starting time of toss was when the curve of accelration started and the time of release
is when it went to 0.

Question 2-3: Can you tell from the data when the IOLab reached its highest point? Why or why not?

No you cannot because the whole time while reases the accerlation was at 0.

Question 2-4: Does the acceleration for dropping the IOLab agree with your prediction? Explain.

Yes, we predicted it would be 0m/s2 when in free fall.

Question 2-5: Does the acceleration after your release for tossing the IOLab up in the air without spin
agree with your prediction? Explain.

yes, we once again predicted it would be 0 m/s when falling.

Slide 11

Accelerometer (400 Hz) Remote 1 Ax Ay Az

20

15

10

5
a (m/s²)

-5

-10

-15

-20
0 2 4 6 8 10 12 14 16 18 20
  Time (s)

INVESTIGATION 3: THE SIMPLE PENDULUM: ANOTHER

EXAMPLE OF SHM
In this Investigation you will explore the motion of a simple pendulum—a mass hanging from a string—
which is a SHM under certain conditions. You will explore what the period of a simple pendulum
depends on.

To complete the next activity you will need:

IOLab
A long string about 2.5 m in length
Smooth tabletop or other flat surface at least 0.5 m long
A meter stick or rod to hang the pendulum from

Activity 3-1: Determining the Period of a Simple Pendulum

1. Set up the IOLab and string as a pendulum as shown in the diagram
on the left, with the string looped around and taped to either side of
the IOLab (in the top left hand side of the diagram) and hung from a
rod such that the y-direction is vertical. (The IOLab can swing in and
out of the page.)

Prediction 3-1: What would be the acceleration, as measured by the

accelerometer, for the IOLab hung in this way at rest? Remember that
the acceleration is a vector quantity. Explain.
 -9.80 m/s2 for the y axis because it will just be the acceleration by gravity.

Prediction 3-2: What would the acceleration, as measured by the accelerometer, be for the IOLab
swinging back and forth—(1) at one end of the swing, (2) at the other end of the swing and (3) at the
bottom of the swing.

The accerleration will be (1) slightly above gravity (2) slighly below gravity (3) gravity.

Test your predictions.

2. Start the pendulum swinging such that it moves about 5 cm forwards and backwards.
Record.
Slide 12
Accelerometer (400 Hz) Remote 1 Ax Ay Az

20
∆t: 4.92341 s
μ: 1.016 m/s² — σ: 0.031 m/s² a: 5.000 m/s s: -0.00 m/s³ (r²: 0.02)
15 μ: 9.704 m/s² — σ: 0.028 m/s² a: 47.779 m/s s: -0.00 m/s³ (r²: 0.00)
μ: -0.441 m/s² — σ: 0.029 m/s² a: -2.169 m/s s: -0.00 m/s³ (r²: 0.00)
10 μ: 0.528 m/s² — σ: 0.13 m/s² a: 2.603 m/s s: 0.00 m/s³ (r²: 0.00)
μ: 10.083 m/s² — σ: 0.56 m/s² a: 49.655 m/s s: -0.04 m/s³ (r²: 0.01)
5 μ: 0.053 m/s² — σ: 0.17 m/s² a: 0.261 m/s s: 0.02 m/s³ (r²: 0.02)
a (m/s²)

-5

-10

-15

-20
0 2 4 6 8 10 12 14 16 18 20
  Time (s)

Activity 3-1: Determining the Period of a Simple Pendulum

Question 3-1: What does the accelerometer measure in the y-direction when the pendulum is at rest?
How does this compare to your prediction? Explain.

It measures 9.8 which is the accerlation from gravity which is what we predicted.

Question 3-2: What does the accelerometer measure when the pendulum reaches one end of its swing
(maximum displacement from equilibrium)? What does it measure when it reaches the other end? Do
these values make sense in terms of your predictions? Explain.

It measures about 10.9 m/s2 and 9.254 m/s2. Yes because at both ends of the oscillation it
is sightly above and below the value of gravity.

Question 3-3: What does the accelerometer measure when the pendulum reaches the bottom of its
swing (zero displacement from equilibrium)? Does this value make sense in terms of your predictions?
Explain.

9.8 m/s2. Yes becasue this is the value of gravity.

 3. Use your graph and the IOLab software to determine the period of the motion of this
pendulum. Explain how you determined the period, and show any calculations.
T1 (s):
T = time/oscillations
T = 4.923 / 10 = 0.4923s

Question 3-4: Also measure the period of the pendulum with a stopwatch. Do the values agree?

Yes with the stopwatch I measured a period of 0.48s. This is super close to our value of
0.4923.
Slide 13

Activity 3-1: Determining the Period of a Simple Pendulum

Prediction 3-3: How would the period of a simple pendulum change as the length of the pendulum is
increased?

A longer string would increase the radius of the movement which would increase the time
of the period.

Prediction 3-4: How would the period of a simple pendulum change as the amplitude of the oscillations
is increased?

The period would be shorter because the pendelum is moving quicker through the
oscillations.
Slide 14
Accelerometer (400 Hz) Remote 1 Ax Ay Az

∆t: 4.66026 s
μ: 9.994 m/s² — σ: 1.2 m/s² a: 46.577 m/s s: -0.10 m/s³ (r²: 0.01)
30 μ: 10.577 m/s² — σ: 2.9 m/s² a: 49.285 m/s s: -0.51 m/s³ (r²: 0.06)
μ: 11.063 m/s² — σ: 2.9 m/s² a: 51.555 m/s s: -0.54 m/s³ (r²: 0.06)
25
20
a (m/s²)

15
10
5
0
-5
-10
0 2 4 6 8 10 12 14 16 18 20
  Time (s)

Activity 3-1: Determining the Period of a Simple Pendulum

Test your predictions.

4. Adjust the length of string from which the IOLab is hanging, and use the IOLab software to
determine the period of the pendulum for a number of different lengths. Measure the length
from your suspension point to the center of gravity of the IOLab, labelled “G” located
between the two buttons. Fill in the table below.
Case Length (m) Period (s)

0.03 0.352
1

0.09 0.410
2

0.14 0.466
3

5. Use the IOLab software to measure the period of the IOLab pendulum for a number of
different amplitudes of oscillation. Be sure that the angular amplitude is under 10° in each
case. Fill in the table below.
Case Amplitudes (m) Period (s)

2
 3

Question 3-5: How does the period of a simple pendulum appear to depend on the length of the
pendulum? How does this compare to your prediction?

The length of the pendelum affects the period because it makes the periods longer if the
pendelum is longer.

Question 3-6: Does the dependence of period on the amplitude support the formula T = 2π√⎯⎯g⎯l ?
Justify your answer using your data.

The amplitude does not effect the period because just the length and mass affect the
period.

Question 3-7: Does the dependence of period on the length support the formula T = 2π√⎯⎯g⎯l ? Justify
your answer using your data.

Yes, because the length showed a gradual increase in the time of the period.