Exploring Simple Harmonic Motion in Physical Pendulums

In summary, the physical pendulum oscillates with simple harmonic motion with angular frequency ω = √(mgd/I).
  • #1
cjavier
17
0
The physical pendulum is an object suspended from some point a distance d from its center of mass. If its moment of inertia about the center of mass is given by:
I= Icm + Md2
where d is the distance from the pivot to the center of mass of the pendulum.

Consider that some odd-shaped physical pendulum of mass M is suspended from some pivot point and displaced through a given angle θ, then released. If the pendulum has a moment of Intertia I about the pivot, then the differential equation describing its subsequent motion is
Id2θ/dt2 = -Mgdsinθ

a) Use the above info to justify that for a sufficiently small angular displacement, the physical pendulum oscillates with simple harmonic motion with angular frequency
ω = √(mgd/I)

SO: I know that I have to follow the argument for a simple pendulum to justify the solution for the physical pendulum. I think that is involves torque, angular acceleration, and/or moments of inertia. I am not sure how to fully justify the angular frequency equation.
 
Physics news on Phys.org
  • #2
What is the differential equation you know for a simple pendulum?
 
  • #3
rude man said:
What is the differential equation you know for a simple pendulum?

I'm confused if this is an actual question or one that is supposed to make me think.

The differential equation for the motion of a pendulum is Id2θ/dt2
 
  • #4
cjavier said:
Id2θ/dt2 = -Mgdsinθ

a) Use the above info to justify that for a sufficiently small angular displacement, the physical pendulum oscillates with simple harmonic motion with angular frequency
ω = √(mgd/I)
The DE above is not SHM. Your first step is to turn it into a DE for SHM by doing an approximation that's valid for small θ. Do you know a suitable approximation?
 
  • #5
cjavier said:
I'm confused if this is an actual question or one that is supposed to make me think.
The latter.

The differential equation for the motion of a pendulum is Id2θ/dt2

That is not an equation. Where's the rest of it?
 

FAQ: Exploring Simple Harmonic Motion in Physical Pendulums

What is a pendulum?

A pendulum is a weight suspended from a pivot point that can swing freely back and forth.

How does a pendulum work?

A pendulum works based on the principles of potential and kinetic energy. When it is pulled to one side, it gains potential energy. As it swings back and forth, this potential energy is converted into kinetic energy and back again.

What factors affect the period of a pendulum?

The period of a pendulum, or the time it takes for one complete swing, can be affected by the length of the pendulum, the weight of the pendulum, and the strength of gravity.

What is the equation for the period of a pendulum?

The equation for the period of a pendulum is T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.

How is a pendulum used in science?

Pendulums are used in science to demonstrate principles of motion and energy, and to measure time. They are also used in various instruments such as clocks, seismometers, and accelerometers.

Back
Top