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Cheng Chek Chee Secondary School

F6 AL Physical Practical (shm-tas)

Title : Simple Harmonic Motion - Mass on a Spring

Objectives:

To study the simple harmonic motion (SHM) of a vertical spring-mass system.

To investigate how the velocity , acceleration and displacement change with time in a SHM

To investigate how the acceleration is related to displacement in a SHM

To investigate how the velocity is related to the displacement in a SHM.

Equipment

Equipment Needed

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Equipment Needed

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Motion Sensor (CI-6742) / 1 / Mass and Hanger Set / 1
Base and Support Rod / 1 / Meter rule / 1
Clamp, right-angle / 1 / Spring, k ~ 2 to 4 N/m / 1

Background

A spring that is hanging vertically from a support with no mass at the end of the spring has a length L (called its rest length). When a mass is added to the spring, its length increases by DL. The equilibrium position of the mass is now a distance L + DL from the spring’s support. What happens when the mass is pulled down a small distance from the equilibrium position?

The spring exerts a restoring force, F = -kx, where x is the distance the spring is displaced from equilibrium and k is the force constant of the spring (also called the ‘spring constant’). The negative sign indicates that the force points opposite to the direction of the displacement of the mass. The restoring force causes the mass to oscillate up and down. And the relationship between the acceleration and displacement is

and where T is period. The negative sign indicates that the acceleration always opposite to the direction of the displacement of the mass.

The period of oscillation depends on the mass and the spring constant.

As the mass oscillates, the energy continually interchanges between kinetic energy and some form of potential energy. If friction is ignored, the total energy of the system remains constant.

Equipment: Stand & Clamp. Masses with hanger (40 grams), 750 interface & notebook motion sensor.

PART I: Computer Setup

1.  Connect the Motion Sensor’s stereo phone plugs into Digital Channels 1 and 2 of the interface. Plug the yellow-banded (pulse) plug into Digital Channel 1 and the second plug (echo) into Digital Channel 2.

2.  Open the document titled “ SHM.ds” in the desktop of the Computer.

3.  Then Save This document of SHM.ds as a new file named “SHM2003Gp8.ds “ For Group 8 and “SHM2003Gp10.ds “ for Gp 10.

4.  You are not allowed to alter the Display of the “SHM.ds” and use the SHM.ds to store any data

PART II: Sensor Calibration and Equipment Setup

• You do not need to calibrate the Motion Sensor.

1.  Using the rod and support stand, the clamp, and the second rod to suspend the spring so that it hangs vertically.

2.  Add mass and hanger and record the equilibrium position of the mass.

3.  Place the Motion Sensor on the floor directly beneath the mass hanger.

4.  Adjust the position of the spring so that the minimum distance from the mass hanger to the Motion Sensor is greater than the Motion Sensor’s minimum distance at the maximum stretch of the spring.

PART III: Data Recording

1.  First, Measure the initial position of the mass by ticking "monitor data" and reading the position display.

2.  Enter the initial Position value in the calculator display and click "accept "

3.  Pull the mass down to stretch the spring about 2 cm. Release the mass. Let it oscillate a few times so the mass hanger will move up-and-down without much side-to-side motion.

4.  Begin recording data by taking the start button.

5.  The plots of the position, velocity and acceleration of the oscillating mass will be displayed. Continue recording for about 6 oscillations.

6.  End data recording.

• The data will appear as ‘Run #1’.

• The displacement-time curve should resemble the plot of a sine function. If it does not, check the alignment between the Motion Sensor and the bottom of the mass hanger at the end of the spring.

You may need to increase the reflecting area of the mass hanger by attaching a circular paper disk (about 2” diameter) to the bottom of the mass hanger.

7.  • To erase a run of data, select the run in the Data list and press the “Delete” key.

Part (iv) Analyzing the Data

A. Position , velocity and acceleration -time graph

1.  Enlarge the window of the position , velocity and acceleration -time graph. Rescale the Graph axes of to fit the data. In DataStudio, click on the ‘Scale to Fit’ button ().

2. 
Study the graphs and Sketch the corresponding Velocity -time and acceleration-time graph in table below.

3.  If the shape of displacement -time graph of the mass is shown as below . draw the velocity-time graph and acceleration accordingly.

Position
time
velocity
time
Acceleration
time

4.  The maximum displacement of the mass is called the amplitude. State whether the magnitude of velocity and acceleration is maximum or minimum when the mass is at the amplitude.

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5.  When the displacement of the mass is zero, the mass is at the equilibrium position. State whether magnitude of the velocity and acceleration is maximum or minimum at the equilibrium position. .

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6.  From the phase relationship of the acceleration-time graph. State the relationship of the direction of acceleration and displacement.

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7.  Use the displacement time graph . Use the computer by Pressing the button of to measure the time for the five consecutive positive peaks and hence find the average period of the shm.

Peak

/ 1 / 2 / 3 / 4 / 5
Time (s)
Period (s)

Average period of oscillation = ______sec .

B.  The Acceleration- displacement graph.

1. Enlarge the Acceleration - displacement graph window and study the acceleration graph. Use the Fit of graph and using linear fit to find the scope of the graph and check the y-intercept of the straight line graph. Sketch the acceleration-displacement graph in fig below

1.  Describe the shape of acceleration-displacement graph of system.

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2. From the above graph deduced the relationship between the

(a)  the magnitude of the acceleration and the magnitude the displacement

(b)  the direction of the acceleration and the direction of the displacement.

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3. Record the slope of the linear fit The slope of the straight line = ______

Calculate value of the period from the slope of the straight line.

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C. Velocity –displacement graph

1. Click and enlarge the v-x graph. Rescale the Graph axes to fit the data. The following window will be appeared.

2.

Sketch the V-X graph
V
X

Press the smart tool button and use the pointer to measure the following

(3)Measure the first three amplitudes of SHM and find the average amplitude of the SHM

Measurement / 1 / 2 / 3
Amplitude

Average Amplitude: ______

(4) Measure the first three speeds of the SHM at the equilibrium position and find the average of the maximum speed

Measurement / 1 / 2 / 3
Speed

Average Maximum speed :______

(5) State the equation relating the velocity V and displacement X of the object and calculate the period of the SHM. From Part C (3) and (4)

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Conclusion :

1. From Data analysis in Part A, B and Part C, State all your findings about the SHM.

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2. Compare the periods obtained in Part (A) , Part (B) and Part (C) and discuss the possible error in this experiment and account for the difference..

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