Physical Pendulum

O

(1)

L

(2)

θ

(3)

CM

(4)

-mgsin(θ)

(5)

mg

(6)

mgcos(θ)

(7)

Physical Pendulum

• Any rigid body that oscillates back and forth under gravity about a pivot point

• But cannot be modeled as a point mass on a string

• The object’s mass distribution must be included in the motion analysis.

A physical pendulum has its mass distributed over a finite area, so it cannot be modeled as a point mass on a string. The motion depends on both the mass and its distribution (moment of inertia).

What makes a pendulum “physical” rather than “simple”?

Restoring Force in a Physical Pendulum

• The force of gravity acting on the object’s center of mass (CM)

• Provides the restoring torque that pulls it back toward its equilibrium position.

Center of Mass (CM) in a Pendulum

• The point where the weight of the object effectively acts.

• In a physical pendulum, the restoring torque is due to the weight acting at this

• Not at a single point like in a simple pendulum.

Because the moment of inertia (I) determines how resistant the object is to rotation. A larger I means the object oscillates more slowly, increasing the period T.

Why does mass distribution affect the period of a physical pendulum?

The force of gravity acts at the CM, creating a restoring torque about the pivot. The farther the CM is from the pivot, the greater the restoring torque.

What is the role of the center of mass (CM) in a physical pendulum’s motion?

τ = −(mgLsin(θ))

Torque on a Physical Pendulum

Iα =−(mgLsin(θ)) → I((d^2)(θ))/(dt^2) = -mgLθ → ((d^2)(θ))/(dt^2) = -(mgL/I)θ

Equation of Motion for a Physical Pendulum

ω = (mgL/I)^1/2

Angular Frequency of a Physical Pendulum

T = 2π((I/mgL)^1/2)

Period of a Physical Pendulum

For a simple pendulum, all the mass is concentrated at one point, so I = mL^2. Substituting this into the physical pendulum formula gives us T = 2π((L/g)^1/2)

Why is the simple pendulum a special case of the physical pendulum?

Increasing L increases the torque and the restoring force, but also increases the distance the mass travels. The combined effect usually increases the period slightly.

How does increasing the distance between the pivot and the center of mass affect the period?

A larger moment of inertia I makes it harder for the pendulum to rotate, so the oscillation slows down and the period increases.

What happens to the period if the pendulum’s moment of inertia increases?

Because when displaced and released, gravity creates a restoring torque about the pivot point, causing oscillatory motion similar to that of a pendulum until friction and air resistance stop it.

Why does any hanging object (like a coffee mug on a hook) behave like a pendulum?