Question

A SIMPLE PENDULUM TAKES 40 SEC TO COMPLETE 10 OSCILLATIONS. WHAT IS THE TIME PERIOD OF THE PENDULUM?

Answer 1

Answer:

time period time taken/ no.of oscillation

40/ 10

4sec

Answer 2

Answer:

Time taken = 40 / 20 = 2 sec. Thus, the time period of the simple pendulum is 2 seconds.

Explanation:

Time taken = 40 / 20 = 2 sec. Thus, the time period of the simple pendulum is 2 seconds.

Time Period of Simple Pendulum

A point mass M is suspended from the end of a light inextensible string whose upper end is fixed to a rigid support. The mass is displaced from its mean position.

Assumptions:

There is negligible friction from the air and the system

The arm of the pendulum does not bend or compress and is massless

The pendulum swings in a perfect plane

Gravity remains constantSimple Pendulum image 2

Time Period of Simple Pendulum Derivation

Using the equation of motion, T – mg cosθ = mv2L

The torque tending to bring the mass to its equilibrium position,

τ = mgL × sinθ = mgsinθ × L = I × α

For small angles of oscillations sin θ ≈ θ,

Therefore, Iα = -mgLθ

α = -(mgLθ)/I

– ω02 θ = -(mgLθ)/I

ω02 = (mgL)/I

ω0 = √(mgL/I)

Using I = ML2, [where I denote the moment of inertia of bob]

we get, ω0 = √(g/L)

Therefore, the time period of a simple pendulum is given by,

T = 2π/ω0 = 2π × √(L/g)

Energy of Simple Pendulum

Potential Energy

The potential energy is given by the basic equation

Potential Energy = mgh

m is the mass of the object

g is the acceleration due to gravity

h is the height of the object

However, the movement of the pendulum is not free fall it is constrained by the rod or string. The height is written in terms of angle θ and length L. Thus, h = L(1 – cos θ)

When θ = 900 the pendulum is at the highest point. Then cos 900  = 0, and h = L . Therefore,

Potential Energy = mgL

When θ = 00, the pendulum is at the lowest point. Then cos 00 = 1. Therefore  h = L (1-1) = 0

Potential Energy = mgL (1-1)  = 0

At all the points in between potential energy is given as mgL (1 – cos θ)

Kinetic Energy

The kinetic energy of the pendulum is given as K.E = (1/2) mv2

m is the mass of the pendulum

v is the velocity of the pendulum

At the highest point, the kinetic energy is zero and it is maximum at the lowest point. However, the total energy is constant as the function of time.

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