Introduction to Dynamics | Sahithyan's Notes

A branch of mechanics, which deals with motion of bodies.

2 parts:

Kinematics
The study of geometric aspects of motion (not referencing the forces)
Kinetics
The analysis of the forces that cause the motion

Kinematics of a Particle

A particle has a mass and negligible size.

When the motion of a particle is along a straight line.

Suppose $x$ is the distance to the particle from a fixed point on its motion path.

When the motion of a particle is along a curve.

Suppose $\overline{r}$ is the position vector of the particle.

Motion (rectangular form)

v_y = \frac{\text{d}y}{\text{d}t}=\dot{y} \;\; \land \;\; v_x = \frac{\text{d}x}{\text{d}t}=\dot{x}

a_y = \frac{\text{d}^2y}{\text{d}t^2}=\ddot{y} \;\; \land \;\; a_x = \frac{\text{d}^2x}{\text{d}t^2}=\ddot{x}

Motion (polar form)

Velocity have a transverse and radial components.

Curvilinear motion has polar acceleration

Acceleration also have a transverse and radial components.

Transverse component
- $a_\theta=r\ddot{\theta}+2\dot{\theta}\dot{r}$
- In vector equation: $\underline{a_\theta} = \underline{\ddot{\theta}}\times \underline{r}+2(\underline{\dot{\theta}}\times \underline{\dot{r}})$
Radial component
- $a_r=\ddot{r}-r\dot{\theta}^2$
- $\underline{a_r} = \underline{\ddot{r}}+ \underline{\dot{\theta}}\times(\underline{\dot{\theta}}\times \underline{r})$

In the acceleration:

Coriolis’ component of acceleration
$2\dot{\theta}\dot{r}$
Centripetal component of acceleration
$-r\dot{\theta}^2 = \underline{\dot{\theta}}\times(\underline{\dot{\theta}}\times\underline{r})$

Unit vectors in transverse and radial directions are denoted by $e_\theta$ and $e_r$ respectively.

\dot{e}_r=\dot{\theta}e_\theta\;\;\land\;\; \dot{e}_\theta = -\dot{\theta}e_r

v= \frac{\text{d}}{\text{d}t}(re_r) = \dot{r}e_r + r\dot{e}_r = \dot{r}e_r+ r\dot{\theta}e_\theta

a= \frac{\text{d}}{\text{d}t} (r\dot{\theta}e_\theta)= (\ddot{r}-r\dot{\theta}^2)e_r+ (r\ddot{\theta}+2\dot{\theta}\dot{r})e_\theta