• Mechanics
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• Kinematics
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• Circular motion

Circular motion is the movement whose trajectory describes a circular arc.

## Introduction to Circular Motion

Angular position $$(\phi)$$
It is analogous to linear position $$s(t)$$.
Unit: $$[\phi(t)]=rad$$.
Angular velocity $$(\omega)$$
It is the linear velocity analog $$v(t)$$.
Unit: $$[\omega(t)] = \frac{rad}{s}$$.
Angular acceleration $$(\alpha)$$
It is analogous to linear acceleration $$a(t)$$.
Unit: $$[\alpha(t)] = \frac{rad}{s^2}$$.
Centripetal acceleration $$(\vec{a}_{c})$$
It is the acceleration that points to the center of the circular trajectory. It keeps the body on the circular path.
Unit: $$[a_{c}]=\frac{m}{s^2}$$ and $$a_{c} = r \omega^2 = \frac{v^2}{r}$$.
Average angular speed $$(\omega_m)$$
$$\omega_m = \frac{\Delta \phi}{\Delta t}$$.
Unit: $$[\omega_m]=\frac{rad}{s}$$
Average angular acceleration $$(\alpha_m)$$
$$\alpha_m = \frac{\Delta \omega}{\Delta t}$$.
Unit: $$[\alpha_m]=\frac{m}{s^2}$$

All units are in the International System ($$IS$$).

We can relate this movement to a linear motion: $$s(t) = r\phi(t)$$, $$v(t) = r\omega(t)$$ and $$a(t) = r\alpha(t)$$.

### Uniform Circular Motion

It is the movement in which the angular velocity is constant and different from zero.

Time related functions in uniform circular motion: \begin{align} \omega(t) &= \omega_0 \notag \\ \phi(t) &= \phi_0 + \omega_0 t \notag \end{align}

To describe a uniform circular motion, these quantities are also used:

Period $$(\tau)$$
Time interval spent by the object to make one complete revolution.
Frequency $$(f)$$ or Greaky letter $$(\nu)$$
Number of repetitions per unit of time. Unit: Hertz $$[f]=\frac{1}{seconds}=Hz$$ .

Period and frequency relation: $$\tau = \frac{1}{f}$$.
The angular velocity relation: $$\omega = \frac{2 \pi}{\tau} = 2 \pi f$$.

### Non-uniform Circular Motion

It is the movement in which the angular acceleration is constant and different from zero. Its equations are analogous to the ones on the uniformly accelerated motion: \begin{align} \phi(t) &= \phi_0 + \omega_0 t + \alpha \frac{t^2}{2} \notag \\ \omega(t) &= \omega_0 + \alpha t \notag \\ \omega^2 &= \omega_0^2 +2 \alpha \Delta \phi \notag \end{align}

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