# Simple Harmonic Motion

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When a mass is suspended and oscillating on a spring of spring constant , its motion with time
can be described by the oscillation equation:
= ⋅ cos( × + ), (1)
where is the displacement of the mass away from the equilibrium position; the oscillation
amplitude; = √
the oscillation angular frequency; and the initial phase of the oscillation.
The oscillation period of such system is
=
2

= 2√

, (2)which can be used to determine the mass of an object if the spring constant is known.
This is indeed the method astronauts in space stations used to monitor their body mass because it
is weightless out there.

Exploration questions:
During the Exploration, conduct the following investigation:
1. using a spring with two different masses to determine the spring constant with the help of Equation (2). A mass scale is available for you to measure the masses of the objects; a
motion sensor and a force sensor can be used to measure the oscillation period.
Please note that some of the following questions may appear in the post-lab quiz.
During the Exploration, conduct some test runs and discuss the following questions with your group member to reach a consensus; if consensus cannot be reached, feel free to discuss the questions with other groups and/or your lab instructor. Some of the questions will be seen in the postlab
quiz.
1. One can use either or both of the motion and force to measure oscillation period, which one do you prefer? Why do you prefer it?
2. How much will oscillation amplitude affect the oscillation period?
3. When you try the spring with different masses, does calculation give you vastly or slightly different values? If they are vastly different, how will you cope with it?
4. The figure to the right shows a position vs time measurement. Your data curve may look similar. You can measure the peak-topeak
time difference to determine the period; or to measure the time difference between several peaks, and calculate the average period. Do the two different methods make a difference? Which method
do you prefer? Once you have answers to all the questions above, and confidently determine spring constant , please proceed to your lab instructor for the Exploration Grade (refer to the grading section below), and the Application section of the next step.

1. A formula of spring constant derived from Equation (2)
2. A formula of mass derived from Equation (2)
4. The determined spring constant value along with relevant calculation.
Please note that points will NOT be marked down if any of the above is wrong; however, points will be deducted based on the following guideline.
-10 -5 No deduction More than half of the materials are missing, illegible and/or poorly organized; results cannot be understood.
Some but less than half of the materials are missing, illegible and/or
poorly organized; efforts have to be made to understand the results.
Everything is legible and well organized; instructors can easily
understand the results.

Exploration notes:
The figure to the right show a suggested equipment setup. In operation, please
1. protect the motion sensor by placing it in a protection cage
2. allow sufficient space between the motion sensor and the mass; anything within 15cm of the sensor will NOT be detected.
3. zero the force sensor when the mass is suspended stationary on the spring.
4. make sure the mass does not swing during oscillation.
5. make sure the oscillation amplitude is not too large, a few centimeters are optimal.

Application:
Ask your lab instructor for an object of unknown mass. You then determine the mass by measuring its oscillation period, and using the spring constant value determined in the previous step. Once you
Please present the object of study, your measurement data, relevant calculation and the determined mass value to your instructor for a grade, which will be determined based on the following guideline:

Measured mass is within ____ of the known value.
≤ ±1.5% ≤ ±2.0% ≤ ±2.5% ≤ ±3.0% ≤ ±3.5% > ±3.5%
Points 20 17 14 11 8 5
Additionally, points will be deducted based on the guideline below.
-10 -5 No deduction More than half of the materials are missing, illegible and/or poorly organized; results cannot be understood.
Some but less than half of the materials are missing, illegible and/or
poorly organized; efforts have to be made to understand the results.
Everything is legible and well organized; instructors can easily understand the results.

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