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Excerpt

A graphical approach to understanding the form of the centripetal acceleration.

Assumptions

We assume that we have uniform circular motion (motion with a constant radius and a constant speed centered at a fixed point in space).

The Diagram

The picture below illustrates the motion, with coordinates chosen so that the angular position at t = 0 is θ = 0.

Image Added

To the right of the motion diagram is a vector diagram that shows the change in the velocity vector. The picture motivates the conclusion that if we take a very small Δt, the change in the velocity approaches:

Latex
Wiki Markup
{excerpt}A graphical approach to understanding the form of the centripetal acceleration.{excerpt}

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h2. Assumptions

We assume that we have _uniform_ circular motion (motion with a constant radius and a constant speed centered at a fixed point in space).

h2. The Diagram

The picture below illustrates the motion, with coordinates chosen so that the angular position at _t_ = 0 is θ = 0.  

!DeltaV.png!

To the right of the motion diagram is a vector diagram that shows the change in the velocity vector.  The picture motivates the conclusion that if we take a very small Δ_t_, the change in the velocity approaches:

{latex}\begin{large}\[ \Delta\vec{v} \rightarrow - v(\Delta \theta)\hat{r}\]\end{large}{latex}

In

...

the

...

infinitesimal

...

limit,

...

this

...

equation

...

becomes:

{
Latex
}\begin{large}\[ \frac{d\vec{v}}{dt} = - v \frac{d\theta}{dt} \hat{r}\]\end{large}{latex}

Using

...

the

...

fact

...

that

...

for

...

uniform

...

circular

...

motion,

{
Latex
}\begin{large}\[ \frac{d\theta}{dt} = \frac{v}{r}\]\end{large}{latex}

we

...

arrive

...

at

...

the

...

form

...

of

...

the

...

centripetal

...

acceleration:

{
Latex
}\begin{large}\[ \vec{a} = \frac{d\vec{v}}{dt}= -\frac{v^{2}}{r} \hat{r}\]\end{large}

Analogy with Gyroscopic Precession

Consider a gyroscope precessing. The angular momentum will trace out a circle as shown below.

Image Added

The similarity to the Δv diagram implies that we can write:

Latex
{latex}

h2. Analogy with Gyroscopic Precession

Consider a gyroscope precessing.  The angular momentum will trace out a circle as shown below.

!DeltaL.png!

The similarity to the Δ_v_ diagram implies that we can write:

{latex}\begin{large}\[ \frac{d\vec{L}}{dt} = L\frac{d\phi}{dt}\hat{\phi} \]\end{large}{latex}

where

...

the

...

derivative

...

of

...

φ with

...

respect

...

to

...

time

...

is

...

the

...

angular

...

frequency

...

of

...

precession,

...

usually

...

written

...

as Ω:

Latex
 Ω:

{latex}\begin{large}\[ \frac{d\phi}{dt} = \Omega\]\end{large}{latex}

h4. Fundamental Relationship for Gyroscopes

With that substitution, we have arrived at the fundamental relation for gyroscopes:

{

Fundamental Relationship for Gyroscopes

With that substitution, we have arrived at the fundamental relation for gyroscopes:

Latex
latex}\begin{large}\[ \frac{d\vec{L}}{dt} = L\Omega\hat{\phi} \]\end{large}{latex}

{note}It is important to note that the relevant model for gyroscopes, [Angular Momentum and External Torque], contains the Law of Change:
\\
{latex}\begin{large}\[ \sum \vec{\tau} = \frac{d\vec{L}}{dt} \]\end{large}{latex}
\\
Often, this relationship will be used in gyroscope problems.{note}