Connexions

You are here: Home » Content » Simple and Compound Pendula

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• Rice Digital Scholarship

This module is included in aLens by: Digital Scholarship at Rice UniversityAs a part of collection: "Waves and Optics"

"This book covers second year Physics at Rice University."

Click the "Rice Digital Scholarship" link to see all content affiliated with them.

Click the tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Simple and Compound Pendula

Module by: Paul Padley. E-mail the author

The Simple Pendulum

Shown is a simple pendulum which has a mass m m that is displaced by an angle θ θ . There is tension ( T T ) in the string which acts from the mass to the anchor point. The weight of the mass is m g m g and the tension in the string is T = m g cos θ T = m g cos θ . There is a tangential restoring force = m g sin θ = m g sin θ . If we approximate that θ θ is small (we have to make this approximation or else we can not solve the problem analytically) then sin θ θ sin θ θ and x = l θ x = l θ . (note that sin θ sin θ is only approximately equal to x l x l because x x is the distance along the x x axis) so that we can write: F = m a = m x ¨ = m g sin θ m g θ m g x l F = m a = m x ¨ = m g sin θ m g θ m g x l or x ¨ + g l x = 0 x ¨ + g l x = 0 (Note that We should immediately recongnize that this is the equation for simple harmonic motion (SHM) with ω = g l . ω = g l .

We could take another approach and use angular momentum to solve the problem. Recall that: L = I ω = I θ ˙ L = I ω = I θ ˙ I = m l 2 . I = m l 2 . Also recall that the torque is the time derivative of the angular momentum so that: τ = r × F = L t l m g θ = I θ ¨ τ = r × F = L t l m g θ = I θ ¨ θ ¨ + g l θ = 0 θ ¨ + g l θ = 0 Again we would recognize that this is simple harmonic motion with ω = g l . ω = g l .

The Compound Pendulum

The compound pendulum is another interesting example of a pendulum that undergoes simple harmonic motion. For an extended body then one uses the center of mass and the moment of inertia. Use the center of mass, the moment of inertia and the Torque (angular force) τ = r × F τ = r × F

τ = r × F I θ ¨ = l m g sin θ l m g θ θ ¨ + l m g I θ = 0 τ = r × F I θ ¨ = l m g sin θ l m g θ θ ¨ + l m g I θ = 0 So again we get SHM now with ω 2 = l m g I ω 2 = l m g I One sees that this formalism can be applied to the simple pendulum (ignore the string and one can consider the ball a point mass). The moment of inertia is m l 2 m l 2 . So we get ω 2 = l m g m l 2 = g l ω 2 = l m g m l 2 = g l which is just what we got before for the simple pendulum. We could write the equation of motion for a simple pendulum as: θ = A e i ( ω t + φ 0 ) θ = A e i ( ω t + φ 0 )

where φ 0 φ 0 is determined by initial conditions.

A discussion of the Pendulum and Simple Harmonic Oscillator can be found at

Content actions

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks