Question

show that the mention of the simole pendulum in simple hormonic motion and describe equation for time period what is seconds pendulum​

Answer 1

Overview:

A simple pendulum is a mechanical arrangement that demonstrates periodic motion. The simple pendulum comprises of a small bob of mass ‘m’ suspended by a thin string secured to a platform at its upper end of length L.

Important Terms

• The oscillatory motion of a simple pendulum: Oscillatory motion is defined as the to and fro motion of the pendulum in a periodic fashion and the centre point of oscillation known as equilibrium position.

• The time period of a simple pendulum: It is defined as the time taken by the pendulum to finish one full oscillation and is denoted by “T”.

• The amplitude of simple pendulum: It is defined as the distance travelled by the pendulum from the equilibrium position to one side.

• Length of a simple pendulum: It is defined as the distance between the point of suspension to the centre of the bob and is denoted by “l”.

Time Period of Simple Pendulum

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

Simple Pendulum

Time Period and Energy of a Simple Pendulum

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,

Therefore, Iα = -mgLθ

α = -(mgLθ)/I

– ω02 θ = -(mgLθ)/I

ω02 = (mgL)/I

ω20 = √(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

From the above figure, the potential energy (PE) of the bob at B with respect to A is,

m×g×h = mg × (L – L cosθ)h = L – L cosθ

= mgL × (1 – cosθ)cosθ = 1 – 2 sin2θ/2

= mgL [1 – (1 – 2 sin2 θ/2)]

For small angles sin (θ/2) ≈ θ/2

= mg × L2(θ/2)2= 1/2 (mgLθ2)

Potential Energy of a simple pendulum is1/2 (mgLθ2).

Kinetic Energy of the Bob:

1/2 (Fω2) = 1/2 (mL2) (dθ/dt)2 [ω = dθ/dt]

=1/2 (ML2) [θ20 ω02 cos2 (ω0t)]

=1/2 (ML2) ω02 [θ02 (1 – sin2 ω0t}]

=1/2 (ML2) × (g/L) × [θ02 – θ2]

∴ Kinetic Energy of a simple pendulum is [1/2 × mgLθ02] – 1/2 mgLθ2θ0 – Amplitude

Mechanical Energy of the Bob:

E = KE + PE= 1/2 mg Lθ02 =constant

The energy of the simple pendulum is conserved and is equal in magnitude to the potential energy at the maximum amplitude.

$bts (♡ω♡ ) ~♪ exo$