Dynamics

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.

Note

When bodies of finite size is of interest, the body might be considered as particles provided motion of the body is characterized by motion of its center of mass and any rotation of the body is neglected.

Rectilinear motion

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. $$

• $\dot{x}$ is its instantaneous velocity.
• $\ddot{x}$ is its instantaneous acceleration.

Curvilinear motion

When the motion of a particle is along a curve.

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

• Instantaneous velocity $v=\frac{\text{d}r}{\text{d}t}$
• Instantaneous speed $|v|=\frac{\text{d}s}{\text{d}t}$
• Instantaneous acceleration $a=\frac{\text{d}v}{\text{d}t}$

Note

Right hand rule is used here to denote the direction of any rotary motions.

2D motion of a particle

Rectangular form

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

Polar form

Velocity have a transverse and radial components.

• Transverse component $v_\theta=\underline{\dot{\theta}}\times \underline{r}$
• Radial component $v_r=\underline{\dot{r}}$

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: $-r\dot{\theta}^2 = \underline{\dot{\theta}}\times(\underline{\dot{\theta}}\times\underline{r})$

Effects of Coriolis’ component

• Objects reflect to the right in the northern hemisphere
• Objects reflect to the left in the southern hemisphere
• Maximum deflections occur at the poles. No deflection at the equator.

Unit vectors

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

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

Velocity

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

Acceleration

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