Physices data repoert (only abstract and result)

Simple Harmonic Oscillator

Equipment:  No special safety equipment is required for this lab.

 Computer with PASCO 850 Universal Interface and PASCO Capstone

 PASCO Motion Sensor

 Vertical Stand with horizontal cross bar.

 Spring kit.

 Mass hanger.

 Set of masses.

Introduction

Imagine a spring that is suspended from a support. When no mass is attached at the end of the

spring, it has a length L (called its rest length). If a mass is added to the spring, its length increases by ∆L. The equilibrium position of the mass is now a distance L + ∆L from the spring’s support

What happens if the mass is pulled down a small distance A from the equilibrium position? The

spring exerts a restoring force, F = -kx, where x is the distance the spring is pulled down and k is the force constant of the spring. The negative sign indicates that the force points opposite of the

direction of the displacement of the mass. The restoring force causes the mass to oscillate or move

up and down within a range of A from the equilibrium. The distance A or maximum displacement

from the equilibrium is called an amplitude of oscillation. The period of oscillation for simple

harmonic motion depends on the mass and the force constant of the spring.

We expect that the frequency of the oscillations will be found from:

𝑓 = 1

2𝜋 √

𝑘

𝑚

Here, 𝑓 is the frequency in hertz (Hz), 𝑘 is the spring constant in N/m, and 𝑚 is the mass attached to the spring, in kg.

Another measure of the oscillations is the period. This is the time for one oscillation. It is:

𝑇 = 1

𝑓 = 2𝜋√

𝑚

𝑘

Note that the amplitude of oscillation is not present in the formulas above and, therefore, it has no

effect on the period and frequency of oscillation.

Objective

 To verify the dependence of a period of a spring-mass system acting as a simple harmonic oscillator on mass, spring constant, and amplitude.

Part #1. Measuring the Spring-Mass System

1. Suspend a green spring from a horizontal support rod and add enough mass to the other end

to stretch the spring so the coils do not touch.

2. Open “Motion Sensor Set Up” file located online next to the lab instructions.

3. Place the Motion Sensor directly under the spring and orient it at 90°.

4. Lightly tap the mass. Let it oscillate a few times so the mass hanger will move up-and-down

without much side-to-side motion. Make sure the coils don’t touch when the mass is at its

highest point. If they do, try to create a more gentle oscillation.

5. While the mass is oscillating, press Record to monitor the position of the mass relative to the

sensor over a period of several oscillations.

6. Rescale the data to fit the Graph window if necessary.

7. Using the Coordinate Tool, measure the time of five consecutive peaks.

8. Use the measured time to calculate the experimental value of the period of oscillations by

calculating the difference between two sequential times.

9. Average the experimental periods and use it to calculate the frequency of oscillation.

Peak Number 1 2 3 4 5

Time (s)

Period (s)

Average Period (s)

Frequency (Hz)

Table 1. Measured and Calculated values for an oscillator build with the ______ spring and

a hanging mass of _____ kg.

Part#2. Period vs. Spring Constant

1. Replace the green spring by each of the spring in the set and repeat steps 3-6 from Part #1 for each spring while keeping the same suspended mass.

2. Do not use the Coordinate Tool, but instead, highlight the smoothers part of each graph and use Best Fit Tool to fit the graph into SIN function.

3. Obtain the value of frequency from the tab that will pop up and record it in Table 2. (For the green spring, you can use the value from Table 1.) Note: if angular frequency, ω is given,

convert it to regular frequency, f in Hz with 𝑓 = 𝜔 2𝜋

.

Color of the spring Green White Yellow Blue Red

Spring Constant (N/m)

Frequency (Hz)

Squared Frequency (Hz2)

Table 2. Oscillation frequencies for several springs, each with the same suspended mass of ___kg.

Since the frequency is proportional to the square root of the spring constant, the frequency squared

should be proportional to the spring constant.

𝑓 = 1

2𝜋 √

𝑘

𝑚

𝑓 2 = 1

4𝜋2𝑚 𝑘

In a graph of 𝑓 2 vs. 𝑘, the slope should be equal to (4𝜋2𝑚)−1

4. Graph Squared Frequency vs. Spring Constant. Fit the plot into a linear function and obtain the equation of the trendline.

5. Calculate (4𝜋2𝑚)−1 and compare it with the slope of the graph.

(4𝜋2𝑚)−1 Slope % Difference

Value

Part #3. Period vs. Mass

1. Keep “most cooperating” spring in the set suspended and repeat steps 3-6 for each spring while changing the suspended mass by increments of your choice.

2. Again, highlight the smoothest part of each graph and use Best Fit Tool to fit the Sine function to the data.

3. Obtain the value of frequency from the tab that will pop up and calculated the period.

Suspended Mass (kg)

Frequency (Hz)

Period (s)

Squared Period (s2)

Table 3. Oscillation frequencies for several masses, each hanging from the ________ spring.

Since the period is proportional to the square root of the mass, the period squared should be

proportional to the mass.

𝑇 = 2𝜋√ 𝑚

𝑘

𝑇2 = 4𝜋2

𝑘 𝑚

In a graph of 𝑇2 vs. 𝑚, the slope should equal 4𝜋2/𝑘.

3. Graph Squared Period vs. Mass and verify that the slope of this graph equals 4𝜋2/𝑘.

Part #4. Frequency vs. Amplitude

How should the period of oscillation change with the change of the amplitude? Once again, perform

an experiment to investigate this. Suspend the spring from the previous experiment and add a mass

of your choice. Throughout this exercise, the mass and spring constant should not change. Repeat

the above procedure starting with a very small amplitude and gradually increasing it. Use the Sine fit

to measure the amplitude as well as the frequencies.

Amplitude (m)

Frequency (Hz)

Table 4. Oscillation frequency at different amplitudes for the _____spring with the mass of ____kg.