> For the complete documentation index, see [llms.txt](https://jamesbrayy.gitbook.io/atar/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://jamesbrayy.gitbook.io/atar/physics/circular-motion.md).

# circular motion

***

## **uniform circular motion**

* uniform circular motion occurs when a body travels at a constant speed and is constantly accelerating towards the centre curvature
* direction is constantly changing and hence velocity is always changing despite no change in speed
  * $$\left| \overrightharpoon{v}\right|=2\pi rf$$
* the direction of velocity is always tangential to the path of motion, while the direction of acceleration is always perpendicular to the velocity of the body

#### ***centripetal force***

* centripetal force is the net force directed towards the centre of curvature that keeps an object moving in uniform circular motion
* as it is a description of the direction the net force is acting in and thus should not be drawn on free-body diagrams

$$\begin{aligned}\overrightharpoon{F}\_c = m \overrightharpoon{a}\_c &= -\frac{m\lvert\overrightharpoon{v}\rvert^2}{r},\hat r \\\[1ex]\left\lvert \overrightharpoon{F}\_c \right\rvert &= \frac{m v^2}{r} \\\[1ex] &= \frac{m\left(2\pi r f\right)^2}{r} \\\[1ex] &= 4\pi^2 m r f^2\end{aligned}$$

#### ***centrifugal force***

* centrifugal force is the sensation that a body feels while moving in a circular path due to being in a non-inertial reference frame
* it is not to be confused with centripetal force or any other force for that matter as it is only a descriptive tool for the physical sensation of centripetal acceleration
* ***constant speed vs constant period***
  * ![](/files/QLeXEgTj8wD9Ffl64fUY)

## **banked curves**

* for a body travelling at a certain speed, it is possible to bank a curve such that no frictional force between the body and the incline to maintain a stable circular path
  * this can be achieved through allowing the horizontal component of the normal force to provide the centripetal force
* for a curve with optimal velocity of $${v}\_{opt}$$:
  * if $${v}*{body}>{v}*{opt}$$, the required $$\left| \overrightharpoon{F}\_c \right|$$ is increased assuming $$m$$ and $$g$$ are constant
    * to maintain the ratio of $${v}^{2}:r$$, the radius of the path of the body must increase
    * this means there must be friction acting between the body and the slope to keep it from sliding up the incline
  * if $${v}*{body}<{v}*{opt}$$, the required $$\left| \overrightharpoon{F}\_c \right|$$ is decreased assuming $$m$$ and $$g$$ are constant
    * to maintain the ratio of $${v}^{2}:r$$, the radius of the path of the body must decrease
    * this means there must be friction acting between the body and the slope to keep it from sliding down the incline

#### ***derivation of*** $$\bm{v\_{opt}}$$

$$
\begin{aligned}
\sum F\_y &= m a\_y = F\_g - F\_{n y} \\\[1ex]
F\_{n y} &= m g \\\[2ex]
\sum F\_x &= m a\_c = F\_{n x} = F\_c \\\[2ex]
\frac{m g}{\cos \theta} &= \frac{m v^2}{r \sin \theta} \\\[1ex]
v^2 &= r g \tan \theta \\\[1ex]
\therefore v\_\mathrm{opt} &= \sqrt{ r g \tan \theta }
\end{aligned}
$$

#### ***example question***

![](/files/Q7vLpyEBwnAcd59vc5ba)

## **vertical centripetal motion**

#### ***upwards circular path***

![](/files/wjLqJU0uiIRTund47xCl)

#### ***downwards circular path***

![](/files/W2GMgTXGvs1ONevIvvWR)

#### ***loop-the-loop***

![](/files/S0VEBfhQ0ZDLO6qUy3Ns)