# Exploring Simple Harmonic Motion: An Experimental Investigation of a Mass-Spring System

Categories: Physics

In this laboratory report, we delve into the principles of Simple Harmonic Motion (SHM) through an experimental investigation involving a mass-spring system. The primary goal was to explore Hooke's law and SHM theory, applying different spring lengths and measurements to analyze the oscillation amplitude and period with the aid of LoggerPro software.

## Objective

The objectiveaof this experimentais to investigateathe motionaof a mass on a springausing Hooke’salaw and the theoryaof simple harmonicamotion. This was done by usingadifferent lengthsaand measurements of the springain relationato the sensor and by usingaLoggerPro to collectadata on the amplitudeaandaperiod.

## Data and Analysis

### Part 1: Equilibrium State

Mass on Spring (kg) Stretch of Spring (Δx) (m)
0 0.758 ± 0.0005
0.05 0.752 ± 0.0005
0.10 0.705 ± 0.0005
0.15 0.656 ± 0.0005
0.20 0.608 ± 0.0005
0.25 0.560 ± 0.0005

### Part 2: Oscillations

Mass (kg) Height Raised (m) Amplitude (m) ± 0.000005 Period T (s) ± 0.005 t₁ (s)
0.2 0.05 0.07834 0.89 0.590
0.2 0.10 0.08122 0.92 0.373
0.3 0.05 0.09443 1.17 0.777
0.3 0.10 0.09469 1.11 0.890

### Analysis 1: Spring Constant (k) Calculation

The slope from the graph was determined as -0.0858 m/N. Thus, the spring constant (k) is calculated as:

N/mk=slope−1=−0.0858−1=11.66N/m

### Analysis 2: Phase Constant Calculation

For a mass of 0.2 kg raised to a height of 0.

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05 m, the phase constant (Φ) was calculated using the formula:

Φ=2π−Tt1

Substituting the given values, we get:

### Spring Constant Calculation (Using ω):

For a mass of 0.2 kg, and angular frequency ω = 7.0597 rad/s, the spring constant (k) is calculated as:

2=9.968 N/mThis value is slightly different from the previous calculation, indicating a deviation of 1.692 N/m.

#### Recorded Values Given Values

m=0.2kg height=0.05m A=0.07834m

C=0.674𝝅

A=0.07816m

C=0.665𝝅

m=0.2kg height=0.10m A=0.08122m

C=0.595𝝅

A=0.08163m

C=0.178𝝅

m=0.3kg height=0.05m A=0.09443m

C=0.336𝝅

A=0.04574m

C=1.634𝝅

m=0.3kg height=0.10m A=0.09469m

C=0.198𝝅

A=0.09232m

C=1.383𝝅

#### Spring Constant (k) Calculation: (using m=0.2kg, ).0597 rad/sω = 7

ω = √ km

m 7.0597) (0.2) .968 N /mk = ω 2 = ( 2 = 9

The valueafor springaconstant calculatedapreviously was 11.66 N/m so it is fairly consistentabut has a 1.692 N/m differenceabetween the two valuesacalculated.

### Conclusion

The experiment conducted provided insightful data on the behavior of a mass-spring system under simple harmonic motion. It was observed that the period of oscillation depends on the spring constant and the mass, but not on the amplitude. The phase constant and spring constant were calculated using the experimental data, which aligns closely with theoretical expectations, barring slight deviations attributed to experimental limitations like friction. The laboratory exercise reinforced the fundamental concepts of SHM and Hooke's law, enhancing our understanding of the dynamics of oscillatory systems.

Updated: Feb 18, 2024