# Simple Harmonic Motion Experiment

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**Kisal Jayakody**

Technical Report

In this experiment, a motion sensor is used to measure the position of an oscillating mass as a function of time. The frequency of oscillations will be obtained by measuring the velocity and acceleration of the oscillations, and fitting the data to a sine function. The dependence of oscillation period on the mass applied and on the spring constant will be studied.

## Introduction

An object oscillating in simple harmonic motion is described by

(1)

where:

y = distance from the equilibrium position at time t

A = amplitude = maximum distance from equilibrium position

f = frequency = number of oscillations per second. An oscillation is one complete back-and-forth motion

ï· = angular frequency of the oscillation = 2ï°f

ïª = initial phase angle

T = The period of the oscillation, .

= the velocity of the mass = .

= the acceleration of the mass = .

## Theory

When a mass hangs from a (massless) spring and oscillates vertically, its period is

where (2)

m = mass hanging from spring

k = spring constant (k = force/elongation)

Squaring both sides,

If the __spring’s mass__ is not negligible this becomes (for a uniform spring)

,

which can be written

.(3)

For a particular spring, this relation of period squared to mass can be written as a linear equation

where and x = mass.

So a graph of T^{2} versus mass should be a straight line with

Slope = (4)

Intercept = (5)

**Pre lab assignment**

- Find the period and the frequency of an object that oscillates 30 times in 44 seconds.

2. In the sample graph, find the value of each of the following quantities; make sure you include proper units!

amplitude

frequency

maximum velocity

maximum acceleration

initial phase angle (of position-time graph)

3.In the sample graph, at t ï€ 0.9 seconds, y = maximum. Explain why

v = 0

a = negative maximum

Also, at t ï€ 1.2 sec., y is at the midpoint of its oscillation. Explain why

v = negative maximum

a = 0

**Apparatus**

Pasco 750 Interface

Motion sensor

Spring, 6 cm by 1.5 cm from Pasco track accessories

Large table clamp, right angle clamp, multi-position pendulum clamp and rods to hold spring and motion sensor (see Figure 1)

50 gram mass holder

50 grams of masses (1x10 gram and 2x20 gram masses)

Meterstick

**Procedure and Analysis for the Simple Harmonic Motion Experiment**

**I.**__Set-up of computer and interface__

- Turn on the Pasco 750 interface
__first__. Notice that the indicator light is on. - Turn on the computer and login.
- Set up the equipment, as shown in Figure 1
- Click on Data Studio, following separate Data Studio instructions.
- Select Motion Sensor.
- Double click on Motion to get to Sensor Properties.
- Under Motion Sensor, increase trigger rate to 25 Hz.
- Under Measurement, select position, and leave velocity and acceleration as selected.
- Click and drag velocity from the Data Window, to the graph icon to create a velocity versus time graph.
- Click and drag acceleration from the Data Window to the bottom of the velocity graph to create an acceleration graph below the velocity graph.
- Click on the lock icon to keep the time axes of the plots locked together.

__Set-up of equipment__

Set-up a desk clamp and rods to hold the spring as in Figure 1. Hang the 50g weight holder from the spring, as shown in Figure 1.

Hold the meterstick in a vertical position next to the weight holder, with the 100cm end touching the floor. Read the position of the bottom of the weight holder; record the total mass on the spring and the position of the mass into an excel spreadsheet. Add 10 grams to the holder and again read and record the mass and position into the excel spreadsheet. Repeat until the
Use the desk clamp and rods to set up the motion sensor as in Figure 1. |
Figure 1. Spring, hanging mass, motion sensor, and miscellaneous rods and clamps for the SHM experiment |

**switch on the sensor for narrow beam** and connect the yellow plug to digital channel 1 of the Pasco interface, and the other plug to channel 2.

**V.**__R____ecording of position-time data during oscillations__

With just the 50 gram holder on the spring, raise or lower the rod holding the

spring until the bottom of the weight holder is about 40 centimeters above the motion sensor. This is done so that the distance from sensor to weight holder will never be less than about 30 centimeters during an oscillation. This is to insure that the motion sensor accurately measures the distance.

Start the weight holder oscillating vertically, about 5 centimeters above and below the equilibrium position. Click on START to begin recording. After a __minimum of 5 oscillations__, click STOP.

**VI.**__Determining the oscillation frequency by a sinusoidal fit__

- Click on Zoom to select the data to be fit. Go to Fit, and select Sine Series Fit. Fit the velocity data, and the acceleration data separately. The data points should form a smooth sine curve. If they don’t, delete the data and record data again. To delete the data, click on run#1 in the experiment set-up window, hit delete, and click on OK.
- The fitted curve should match the data.
- Into a second excel spreadsheet, record the mass on the spring, the amplitude of the velocity, of the acceleration, the frequency of the velocity, and the acceleration.
- Print out a few representative graphs to be included with your laboratory report.

5. Increase the hanging mass to 60g (total) and again adjust the spring support so that the mass hanger is about 50 cm above the motion sensor. Repeat V and VI.

6. Repeat the above steps for a __total__ mass of 70, 80, 90, and 100 grams

7. Finally, disassemble the apparatus and measure the mass of the spring on a balance.

**VII.**__Calculations__

**EQUATIONS**:-

09/20/161

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