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.
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.)
To measure the work done in stretching a spring in the following activity, you will need:
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).
Force (N) | Position (m) |
Spring constant from parametric plot:
Spring constant from Excel:
Question 1-2: Was the force exerted by the spring proportional to the displacement of the spring?
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.
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.)
Prediction 1-2: As the IOLab oscillates up and down, how will the period change as the amplitude is changed?
To test your predictions in the following activities, you will need:
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.
T1 (seconds):
T2 (seconds):
Question 1-4: For simple harmonic motion of a mass on a spring, the period can be calculated from . Using the spring constant you measured previously and the mass of the IOLab you also measured previously, calculate the period of oscillations. show your calculation.
T (seconds):
How does this calculated period compare to the experimentally determined one?
Question 1-5: Does the period appear to depend on the amplitude of the oscillations? Explain.
Question 1-6: How do the directions of the force and acceleration compare? Can you explain why?
Question 1-7: How do the magnitudes of the force and acceleration compare? Can you explain why?
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?
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?
The following activities should help you to see whether your predictions make sense.
ax (m/s²):
ay (m/s²):
az (m/s²):
ax (m/s²):
ay (m/s²):
az (m/s²):
What did you observe while turning the IOLab? Did it matter how fast you turned it or how you turned it?
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.
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.)
Prediction 2-2: What acceleration would you measure when you toss the IOLab up in the air so that it does not spin? Explain.
The following activities should help you to see whether your predictions make sense.
Question 2-1: How do you know?
Question 2-2: How can you tell?
Question 2-3: Can you tell from the data when the IOLab reached its highest point? Why or why not?
Question 2-4: Does the acceleration for dropping the IOLab agree with your prediction? Explain.
Question 2-5: Does the acceleration after your release for tossing the IOLab up in the air without spin agree with your prediction? Explain.
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:
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.
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.
Test your predictions.
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.
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.
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.
T1 (s):
Question 3-4: Also measure the period of the pendulum with a stopwatch. Do the values agree?
Prediction 3-3: How would the period of a simple pendulum change as the length of the pendulum is increased?
Prediction 3-4: How would the period of a simple pendulum change as the amplitude of the oscillations is increased?
Test your predictions.
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Case | Amplitudes (m) | Period (s) |
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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?
Question 3-6: Does the dependence of period on the amplitude support the formula ? Justify your answer using your data.
Question 3-7: Does the dependence of period on the length support the formula ? Justify your answer using your data.
Please remember to edit the report (insert your name - and if necessary your partners), export the report and submit it on D2L.
Now do the homework associated with this lab.