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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.
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:
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| Wiki Markup |
{excerpt}A graphical approach to understanding the form of the centripetal acceleration.{excerpt} ||Page Contents|| |{toc:indent=10px}| 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
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equation
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becomes:
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}\begin{large}\[ \frac{d\vec{v}}{dt} = - v \frac{d\theta}{dt} \hat{r}\]\end{large}{latex} |
Using
...
the
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fact
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that
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for
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uniform
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circular
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motion,
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}\begin{large}\[ \frac{d\theta}{dt} = \frac{v}{r}\]\end{large}{latex} |
we
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arrive
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at
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the
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form
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of
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the
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centripetal
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acceleration:
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}\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.
The similarity to the Δv diagram implies that we can write:
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{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
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the
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derivative
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of
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φ with
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respect
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to
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time
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is
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the
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angular
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frequency
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of
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precession,
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usually
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written
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as Ω:
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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:
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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} |