8.01SC | Fall 2016 | Undergraduate

# Classical Mechanics

Week 3: Circular Motion

## 10.1 Circular Motion – Acceleration

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Acceleration in Circular Motion: $$\vec{a} = r\frac{d^2\theta}{dt^2}\hat{\theta} + r(\frac{d\theta}{dt})^2(-\hat{r})$$

For circular motion, show that $$\frac{d \hat{\theta}}{dt} = -\frac{d \theta}{dt}\hat{r}$$

We know that $$\hat{\theta }$$ changes direction as we go around a circle, and so if a particle is undergoing circular motion, we expect $$\frac{d\hat{\theta }}{dt}$$ to be non-zero. To figure out exactly what it is, let us write $$\hat{\theta }$$ in terms of $$\hat{i}$$ and $$\hat{j}$$ for an arbitrary $$\theta$$. \begin{align} \hat{\theta} = -\sin(\theta) \hat{i} + \cos(\theta) \hat{j} \end{align}

Since $$\hat{i}$$ and $$\hat{j}$$ are constant anywhere in space, they are not time-dependent, so now the derivative becomes much more clear. Using the chain rule, we have:

\begin{align} \frac{d\hat{\theta}}{dt} = -\cos(\theta) \frac{d\theta}{dt} \hat{i} - \sin(\theta) \frac{d\theta}{dt} \hat{j} = - \frac{d\theta}{dt} \left(\cos(\theta) \hat{i} + \sin(\theta)\hat{j} \right) \end{align}

We know that $$\left(\cos(\theta) \hat{i} + \sin(\theta)\hat{j} \right) = \hat{r}$$, so we can now simply write this derivative as:

\begin{align} \frac{d\hat{\theta}}{dt} = - \frac{d\theta}{dt} \hat{r} \end{align}

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