Derivation Of Expression For Its Time Period

A simple pendulum is a mechanical plan that shows periodic motion. The simple pendulum involves a little bob of mass ‘m’ suspended by a slim string to a stage at its upper finish of length L.

The simple pendulum is a mechanical framework that influences or moves in an oscillatory motion. This motion happens in an upward plane and is fundamentally determined by gravitational energy. Interestingly, the bob that is suspended toward the finish of a string is exceptionally light. We can say that it is even massless. The time of a simple pendulum can be made stretched out by expanding the length string, while at the same time taking the estimations from the mark of suspension to the center of the bob. It should be noticed that if the mass of the bob is transformed, then the period will stay unaltered. The period is impacted essentially by the place of the pendulum according to the earth as the strength of the gravitational field is not uniform all over.

Simple harmonic motion involves a simple repetitive motion of back and forth, a central position, and the furthermost displacement on either side of the central position is the same. Each vibration is completed in the same time period. The force of motion is directed towards the central position at all times during the motion and the force is directly proportional to the distance from the central position.

F = -kx

In this equation, the negative sign (-) shows that the force is directed towards the opposite direction. The term ‘k’ in this equation is a constant term called the force constant. The unit of this term ‘k’ is Newton per metre.

Let the mass of the spring be m. In that case, the acceleration (a) will be:

a=Fm

a=-kxm

a =–𝜔2x

By comparing coefficients, km=  𝜔2

The time period is the time required by the object to complete one oscillation. The frequency of simple harmonic motion is the total number of oscillations taken by the object per unit of time. Therefore, we can represent the frequency as:

f = 1/T

In the above equations:

• a = acceleration 

• T= time period

• F = force

• f = frequency 

• m = mass

• 𝜔 = angular frequency

• k = force constant 

Key Terms

1. Frequency: The number of oscillations completed per unit of time is the frequency of the simple harmonic motion being performed by the particle. The SI unit of frequency is Hertz. 

2. Amplitude: The magnitude of maximum displacement of a particle from its mean position during simple harmonic motion is called its amplitude. 

3. Time period: The time in which the particle completes one oscillation is called its time period. 

4. Mean position of equilibrium: It is the position of the particle during the motion when the net force acting on it is zero.

Frequency and Period of Simple Harmonic Motion

The period or the time taken to complete one oscillation and the frequency or the number of oscillations per unit time of simple harmonic motion can be calculated because it is a periodic motion. Two experiments can be used for this purpose:

1. Pendulum: A mass m is attached to a pendulum of length L. It will oscillate in a period T. The following formula describes the motion:

T = 2π√(L/g), where g is the gravitational acceleration 

2. Mass on a spring: A mass m is attached to a spring with a spring constant k. The period t is described by the following formula:

T = 2π√(m/k)

In the case of the pendulum, the time period is independent of the mass, whereas in the mass spring, the time period depends on the mass.

Time Period of Simple Pendulum Derivation

Utilising the condition of motion, T – mg cosθ = mv 2 L

The force has a tendency to carry the mass to its harmonious position,

τ = mgL × sinθ = mgsinθ × L = I × α

For little points of motions sin θ ≈ θ,

Thus, Iα = – mgLθ

α = -(mgLθ)/I

– ω02 θ = – (mgLθ)/I

ω02= (mgL)/I

ω0 = √(mgL/I)

Utilising I = ML2, [the snapshot of idleness of bob]

we get, ω0 = √(g/L) 

Thus, the time span of a simple pendulum is given by

T = 2π/ω0 = 2π × √(L/g)

Conclusion

Now and again, people think that a simple pendulum’s period relies upon the displacement or the mass. Expanding the adequacy means that there is a bigger distance to travel; however, the reestablishing power likewise builds, which relatively builds the speed increase. This implies the mass can travel a greater distance at a more prominent speed. These attributes drop one another, so adequacy has no impact on the period. The pendulum’s inactivity opposes the coarse adjustment, but on the other hand, it’s the wellspring of the reestablishing power. Therefore, the mass counteracts as well.