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Determination of the time period for oscillation of a simple pendulum by using Simple Harmonic Motion(SHM) method and hence calculate the value of acceleration due to gravity(g)

Theory

Concept
Many things in nature wiggle in a periodic fashion; they vibrate. One example is a simple pendulum. If we suspend a mass at the end of a string, we have a simple pendulum. The to-and-fro motion represents periodic motion used in the past to control grandfather and cuckoo clocks. Such oscillatory motion depends on the size of the arc through which it swings. Another factor affects the period. Called simple harmonic motion, it was Galileo who first observed that the time a pendulum takes to swing back and forth through small distances depends only on the length of the pendulum. The time of this to-and-fro motion, called the period, does not depend on the mass of the pendulum. Involved in the period is the acceleration due to gravity (g), which on Earth is 9.8 m/s². A simple pendulum consists of a massless and inelastic thread fixed at one end to a rigid support, with a small bob of mass 𝑚 m suspended at the other. Let 𝐿 L be the effective length. When the bob is slightly displaced and released, it oscillates about its equilibrium position. This is the theory for determining the time period of a simple pendulum using the simple harmonic motion method and hence calculating the acceleration due to gravity

Concept

Many things in nature wiggle in a periodic fashion; they vibrate. One example is a simple pendulum. If we suspend a mass at the end of a string, we have a simple pendulum. The to-and-fro motion represents periodic motion used in the past to control grandfather and cuckoo clocks. Such oscillatory motion depends on the size of the arc through which it swings.

Another factor affects the period. Called simple harmonic motion, it was Galileo who first observed that the time a pendulum takes to swing back and forth through small distances depends only on the length of the pendulum. The time of this to-and-fro motion, called the period, does not depend on the mass of the pendulum. Involved in the period is the acceleration due to gravity (g), which on Earth is 9.8 m/s².

A simple pendulum consists of a massless and inelastic thread fixed at one end to a rigid support, with a small bob of mass 𝑚 m suspended at the other. Let 𝐿 L be the effective length. When the bob is slightly displaced and released, it oscillates about its equilibrium position.

This is the theory for determining the time period of a simple pendulum using the simple harmonic motion method and hence calculating the acceleration due to gravity

Table & Data Collection

Table-I

No of obs.	Linear Scale Reading(mm)	Circular Scale Divisions	Least Count(mm)	Value of Circular Scale(mm)	Total Reading(mm)	Diameter(d)(mm)	Mean Radius, r=d/2(mm)
1
2
3
4

Table-II

No of obs.	No. of Oscillations	Total time, t(sec)	Time period, T(sec)	Mean time period, T(sec)
1
2	15
3
4
5	20
6
7
8	25
9

Calculations & Results

Calculation
Effective Length, L = l + r (m)
Determining value of simple pendulum using value of g, can be calvulated using equation: g = 4π² ^L⁄_T²
Error: E = \| ^{g - g'} ⁄ _g \| x 100%
Results: Acceleration due to gravity = ms^-2 Error = %

Discussion

Point	Explanation
1	The apparatus was set up with a light string and a spherical bob fixed to a rigid support.
2	The length of the pendulum was measured carefully from the point of suspension to the center of the bob in table-I.
3	The bob was displaced through a small angle (>5 degrees) to maintain simple harmonic motion.
4	The pendulum was releases gently without any push to avoid irregular motion.
5	Tie for 10, 15 and 20 oscillations was measured using a stopwatch for better accuracy in table-II.
6	The experimented was repeated for 3 times in each case, and an average time period was calculated.
7	Effective length (L), was found along with g'.
8	Care was taken to avoid parallel error while measuring length and human error in timing.
9	Yet we calculated errors and found in percentage.
10	Minor errors occured due to air resistance, large amplitude or small errors in stopwatch.