Time Period of Simple Pendulum Derivation

The formula of time period of simple pendulum derivation can be used to calculate the equilibrium, equations of motion, and other factors related to it. The equation in the article gives the derivation of the formula of time period of a simple pendulum. 

It has been assumed that the string that suspends a simple pendulum does not change length during a swing – which is true for small oscillations. Let us understand the derivation in the article by using assumptions to understand the equilibrium state of the pendulum oscillation and the Time period of the pendulum in that equilibrium state.

Simple Pendulum

A simple pendulum is a physical system that consists of a pendulum coupled to the ground via some massless string or rod. The simplest mechanical systems are those that can be modelled as simple harmonic oscillators, with the mass and stiffness of the spring and mass being related by some constant and having different values in opposite directions. In our case, when the mass of the pendulum bob is concentrating at its lowest point, we have to maintain it in equilibrium.

The Energy of Simple Pendulum

For a pendulum, then the total energy at any given point during an oscillation is the sum of kinetic energy and potential energy of the system. The pendulum has a certain amplitude, A, which is the maximum length reached by the pendulum while oscillating. If we have a negative amplitude, then the total energy of the pendulum system will be positive. On the contrary, if we have a positive amplitude, it will be negative.

Uses of a Simple Pendulum

The simple pendulum is used in a number of applications, ranging from the study of physical phenomena at the fundamental level to control and instrumentation. For example, it is useful in studying eddy current losses in electrical power transmission lines. It can be used to estimate the loss coefficient because it exhibits a small degree of damping with time; and because it can be easily studied by means of experimental techniques.

It is also widely used for test purposes in areas such as mechanical engineering and electronics. In the Mechanical engineering field, it is used to test the behaviour of vehicles and machines. In electronics, it is used for determining the stability of miniature devices, semiconductor chips, and batteries.

Equilibrium

In the equilibrium state of the pendulum, the centre of mass is stationary with time period T. The centre of mass of a simple pendulum has been shown to rotate around an axis that passes through its centre of gravity with period T. 

Assuming that the angle between the string and the vertical is called “θ.” If there

is no force acting on the system, then θ = 0, and there is no torque around

the point of suspension.

The sum of all torques due to externally applied forces must be zero. 

When there is no external force exerted on a system, it is said to be in “equilibrium.”

The Time Period of Simple Pendulum Derivation

The derivation of the formula of time period of a simple pendulum depends upon three assumptions; these assumptions are listed below: 

• The string attached to the bob does not change length during oscillation. 
• The gravitational acceleration (g) on earth is 9.8 m/s which has been chosen as the independent variable.
• The length of the pendulum bob is small enough for the discussion.

Time Period of Simple Pendulum Derivation:

According to the equations of motion,

E = mv² + l mgcosθ l

The system tries to attain an equilibrium state, and the Force acting on the mass is Torque.

Torque tends to bring the system to equilibrium,

τ = mgl * sinθ = mgsinθ * l = I * α

Let sinθ ≈ θ, (small oscillations)

I * α = -mglθ 

α = -(mglθ)/I

ω02 θ = -(mglθ)/I

ω02 = (mgl)/I

ω0 = √(mgl/I)

I = ml²

ω0 = √(g/l)

Hence, T =  2π/ω0

T = 2π*√(l/g)

Here, I = Moment of Inertia of Bob

g = acceleration due to gravity

l = Length of the arm of the pendulum

T = Time period of the simple pendulum

m = mass of the Bob

The equation of motion depends upon the gravitational acceleration (g), the mass of the object (m), and the length of string (l). So we have to find g, m, and l. From the formula of time period, we understood that the equation of motion depends upon g, m, and l. 

Conclusion

The time period of the simple pendulum depends upon its length and gravitational acceleration. It is equal to twice the square root of length divided by gravity.

For a physical system, the time period depends on its amplitude, length, and mass. It equals twice the square root of length divided by twice the square root of amplitude.