Acceleration of Particle moving in Circle

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Theorem

Let $P$ be a particle moving in a circular path $C$.

Then the acceleration of $P$, denoted by $\mathbf a \in \R$, is given as:

$\mathbf a = -\dfrac {\size {\mathbf v}^2 \mathbf r} {\size {\mathbf r}^2}$

where:

$\mathbf v$ is the instantaneous velocity of $P$
$\mathbf r$ is the vector whose magnitude equals the length of the radius of $C$ and whose direction is from the center of $C$ to $P$
$\size {\, \cdot \,}$ denotes the magnitude of a vector.


Proof


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We use Lagrangian mechanics to derive the result.

Let $\mathbf r = \map {\mathbf r} t$ be a function of time $t \in \R_{\ge 0}$ .

Let $R = \size r \in \R_{>0}$ be the constant radius of the circle $C$.

The constraint can then be written as:

$\forall t: \map f {\mathbf r} := \size {\mathbf r}^2 - R^2 = 0$


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The Principle of Stationary Action states that the first variation of the action $S$ must vanish.

$\ds \delta S = \delta \int L \rd t = 0$


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The Lagrangian $L$ is given by the kinetic energy alone, because the constraint force is handled implicitly by the constraint, and there are no other forces acting on $P$.

$L = \dfrac 1 2 m \size {\mathbf v}^2$

To incorporate the constraint, we use a scalar Lagrange multiplier $\lambda$.

Because we are dealing with functionals, and not functions, and the constraint is local and not global, $\lambda$ is also a function of time $t$, as is $\mathbf r$:

$\lambda = \map \lambda t$

The augmented action $S^+$ is given by:

$\ds S^+ = S + \int \lambda f \d t$


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We have that:

$\dfrac {\d f} {\d \mathbf r} = 2 \mathbf r$

Then:

$\ds \delta S^+ = \int \paren {-\map {\dfrac \d {\d t} } {m \mathbf v} + \map \lambda {2 \mathbf r} } \cdot \delta \mathbf r \rd t + \int f \delta \lambda \rd t = 0$

This means that both the terms in the brackets as well as $f$ must vanish.


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We eliminate $\lambda$ from the term in the brackets by taking its scalar product with $\mathbf r$ and simplifying $\map {\dfrac \d {\d t} } {m \mathbf v}$ to $m \mathbf a$:

$-\map {\dfrac \d {\d t} } {m \mathbf v} \cdot \mathbf r + \map \lambda {2 \mathbf r} \cdot \mathbf r = 0$

This gives:

$2 \lambda = \dfrac {m \mathbf a \cdot \mathbf r} {R^2}$

We must replace $\mathbf a \cdot \mathbf r$.

We take the second derivative with respect to $t$ of the constraint, which must vanish too:


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$\dfrac {\d^2 f} {\d t^2} = 2 \map {\dfrac \d {\d t} } {\mathbf r \cdot \mathbf v} = 2 \paren {\mathbf a \cdot \mathbf r + \size {\mathbf v}^2} = 0$

So:

$2 \lambda = \dfrac {-m \size {\mathbf v}^2} {R^2}$

Substituting $2 \lambda$ in the term with the big brackets yields:


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$-m \mathbf a - \dfrac {m \size {\mathbf v}^2} {R^2} \mathbf r = 0$

Division by $m$ completes the proof.


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$\blacksquare$


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Sources

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