Torsional Pendulum

Θ

Symbol that represents some small maximum angle when the body of the torsional pendulum is twisted.

The body oscillates between (θ = +Θ) and (θ = -Θ)

What happens when the body of the torsional pendulum is twisted to some maximum angle and released from the rest?

By the shearing of the string or wire.

How is the restoring torque in the torsional pendulum is supplied?

κ

Symbol that represents the torsion constant of the wire or string.

τ = -κθ

Formula for the restoring torque of the Torsional pendulum

That in a torsional pendulum a restoring torque acts in the opposite direction to increasing angular displacement.

What does negative sign inside the Restoring Torque equation says about the Torsional pendulum?

I × ((d^2)(θ))/(dt^2)) = -κθ

Rewriting the restoring torque equation where the net torque is equal to the moment of inertia times the angular acceleration.

((d^2)(θ))/(dt^2)) = -(κ/I)θ

Equation for d²θ/dt²

The second time derivative of the position (in this case, the angle) equals a negative constant times the position.

Explain d²θ/dt² = -(κ/I)θ

((dt^2)(x))/(dt^2) = -(k/m)x

SHM equation of motion

T = (2π)((m/k)^1/2)

SHM formula for period

T = 2π((I/κ)^1/2)

Formula of Period in a Torsional Pendulum

κ = Ν × m = (kg × (m/(s^2))m) = kg((m^2)/(s^2))

Unit of κ

I = kg × m^2

Unit for the moment of inertial

dx

(1)

x

(2)

L

(3)

M

(4)

λ = dm/dx = M/L

(5)

I_CM = ∫(x^2)dm = ∫(from -[L/2] to +[L/2]) x^2λdx = λ[(x^3)/3]from -L/2 to +L/2 = λ((2L^3)/24) = (M/L) (2L^3)/24) = (1/12)M(L^2)

Complete Moment of Inertia equation.

shorter

The larger the torsion constant, the _______ the period.