Circular motion

Uniform circular motion

Radial acceleration:

$$a_{rad} = \frac{v^2}{r}$$

Period:

$$T = \frac{2\pi r}{v} = 2\pi\sqrt{\frac{r}{a_{rad}}}$$

Object inside rotating cylinder

$$\begin{align*} n &= \frac{mv^2}{r} - mg && \\ v &\gt \sqrt{rg} && \text{if } n \gt 0 \end{align*}$$

• $n$: normal force (at the top of the cylinder) acting on the object
• if $n \le 0$ (i.e. $v \le \sqrt{rg}$) then the object loses contact with the surface

Conical pendulum

$$\begin{align*} a_{rad} &= g\tan\theta \\ T &= 2\pi\sqrt{\frac{L\cos\theta}{g}} \end{align*}$$

Car rounding curve without skidding

Flat curve

$$\begin{align*} \frac{mv^2}{r} &= \mu_s mg \\ \therefore v_{max} &= \sqrt{\mu_s gr} \end{align*}$$

Banked curve

$$\begin{align*} \sum F_x &= \frac{mv^2}{r} = n\sin\theta + \mu_s n\cos\theta && (1) \\ \sum F_y &= 0 = n\cos\theta - \mu_s n\sin\theta - mg && (2) \\ \therefore v_{max} &= \sqrt{gr\frac{\tan\theta + \mu_s}{1 - \mu_s \tan\theta}} \\ &\ge (v_{flat})_{max} \end{align*}$$