Exploring Simple Harmonic Motion: Investigations into Pendulum Dynamics and Spring Mechanics

Introduction

Simple harmonic motion, often abbreviated as SHM, describes the rhythmic back-and-forth movement of an object around its central or equilibrium position. This phenomenon is commonly observed in various natural and man-made systems, such as swings, swaying trees, pendulums, bobbing boats, and even the swinging motion of one's arm while walking. In SHM, the acceleration of the object is always directed towards its equilibrium position and is directly proportional to the displacement from that position.

For SHM to occur, there must be a restoring force acting on the object, pulling it back towards its equilibrium position.

This force is proportional to the distance the object is displaced from equilibrium. Mathematically, this relationship is expressed as

F = -kx

where $F$ represents the force, $k$ is a constant known as the spring constant, and $x$ denotes the displacement from equilibrium. The negative sign indicates that the force always opposes the direction of displacement, ensuring that the object returns to its equilibrium position.

This fundamental principle governs the motion of objects undergoing simple harmonic motion and provides a basis for understanding various oscillatory phenomena in nature and engineering systems.

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Objectives

Experiment 1 aims to assess the vertical motion of a mass suspended from a spring, determining its adherence to the principles of Simple Harmonic Motion (SHM). Through meticulous measurement and analysis, this experiment seeks to validate whether the displacement of the spring aligns with the expected behavior described by the equation $F=-kx$. By varying the masses attached to the spring and observing any corresponding changes in the period of motion, we endeavor to ascertain the consistency of the period​​, characteristic of SHM.

In Experiment 2, the objective is to accurately determine the acceleration due to gravity utilizing a simple pendulum setup. This investigation involves the precise measurement of the pendulum's oscillations and the subsequent calculation of based on the obtained data. Through this experimental inquiry, we aim to deepen our understanding of gravitational forces and validate theoretical predictions within the context of pendulum motion.

Theory

Experiment 1

The illustration above depicts a mass suspended from a spring, showcasing the equilibrium state where the gravitational force acting on the mass is precisely counterbalanced by the force exerted by the spring. In this equilibrium position, the mass remains stationary. However, any deviation from this position results in the emergence of a restoring force, directing the mass back towards its equilibrium point. When the mass moves below this equilibrium point, the spring exerts an upward force, while gravity pulls it downwards when it surpasses the equilibrium. Consequently, the net force acting on the mass functions as a restoring force, perpetually striving to return the mass to its stable position.

Experiment 1 delves into the exploration of Hooke's Law, a fundamental principle in the realm of spring mechanics. Hooke's Law posits that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.

Experiment 2

A simple pendulum can be conceptualized as a point mass suspended from a massless string, allowed to swing back and forth about a fixed point. It typically consists of a weighty bob hung from a string of negligible mass. When the bob is displaced from its equilibrium position and then released, it undergoes oscillations around this central point, resulting in periodic motion. The pivotal question arises: does this motion exhibit characteristics of simple harmonic motion?

A fundamental criterion for simple harmonic motion dictates that the restoring force must be directly proportional to the displacement (denoted as $x$) from equilibrium and act in the opposite direction of this displacement. However, in the case of a pendulum, the trajectory of the bob traces a circular path with a radius equal to the length of the pendulum ($L$). Here, the displacement $x$ refers to the distance along this curved path measured from the equilibrium point.

The standard formula for simple harmonic motion, $F=-kx.$

Apparatus

Experiment 1

• Experiment board
• Spring balance
• Mass hanger
• Masses
• Stopwatch

Experiment 2

• Plum bob
• Stopwatch
• Non-extensible string
• Retort stand
• Meter stick
• Clamps
• Wooden blocks

Procedures

Experiment 1

Measuring the value of the spring constant, \( k \):

1. Measure the hanger without the load.
2. Hang masses of different values and measure the displacement of the spring for each.
3. Plot a graph of force versus displacement to determine the spring constant, \( k \).

Measuring the period:

1. Set up the equipment with a fixed mass.
2. Initiate oscillations and measure the time for a specified number of oscillations.
3. Repeat the measurements for different masses and calculate the period using the formula.

Experiment 2

Measuring the period of a simple pendulum:

1. Set up the pendulum and measure the length.
2. Record the time for a specific number of oscillations.
3. Repeat the measurements for different pendulum lengths.
4. Calculate the period and analyze the relationship between period and length.

Results

Experiment 1

For Experiment 1, the following results were obtained:

Experiment 2

For Experiment 2, the following results were obtained:

Discussion

EXPERIMENT 1

In experiment 1, we varied the mass which were; 0.125 kg, 0.175 kg, 0.225 kg. The oscillations for all the mass were constants which was 10. For 0.125 kg, the time in second measured for all the oscillations were 3 second. Next, we change the mass to 0.175 kg and the time measured for the oscillations were 4 seconds.  Last, for the mass of 0.225 kg, the time measured were 5 seconds. The average period was gained by plus all the time measured and divides by 5 oscillations. The period was calculated using the formula T = 2π√M (kg)/k and the value were 0.360 s for 0.125 kg, 0.426 s for 0.175 kg, 0.483 s for 0.225 kg respectively. The mass is directly proportional to the period.                                    

EXPERIMENT 2

For experiment 2, the length of the pendulum was change from 0.13 m, 0.17 m and 0.21 m.The time was measured twice so that the average time was gained. To get the time for a single oscillations, the average time was divided by 30 for 30 oscillations. From the data, we can plot the graph of T² versus the length of the pendulum. From the graph, we can determine the slope of the graph which was 15.5 s²/m. using the slope, we can calculate the value for the gravity. From the graph, we can conclude that the length of pendulum is directly proportional to T². 

Conclusion

Experiment 1 demonstrates that the motion of a simple pendulum aligns with the characteristics of simple harmonic motion, thus confirming its nature as such. Variations in mass yield corresponding changes in the calculated period, indicating a direct relationship between mass and period length. Specifically, increasing mass results in a higher period value.

In Experiment 2, a graph plotting 2T2 against the length of the pendulum ($L$) was generated. The calculated percent difference between the theoretical and experimental values of gravity was found to be 74.01%. Analysis of the graph reveals that altering the pendulum's length directly influences its period, with shorter strings yielding shorter periods.

Several factors contributed to experimental errors, including friction on the paper tape attached to the pendulum mass, air resistance, and inaccuracies in the ticker-timer. These errors likely impacted the precision of the results obtained.