What is Kinematics?
Kinematics refers to the study of motion, disregarding the causes of motion. As such, kinematics is concerned primarily with the position, velocity, and acceleration of objects.
Let’s quickly define some notation and terminology.
- \bm{\overrightarrow{r}} is the position of particle from an origin point, O to a point A.
- \bm{\overrightarrow{r}^\prime} is the position of particle from an origin point, O to a point A^\prime
- \Delta\bm{\overrightarrow{r}} denotes displacement of a particle from a point A to A^\prime. \Delta\bm{\overrightarrow{r}}=\bm{\overrightarrow{r}^\prime}-\bm{\overrightarrow{r}}.
- A dot above a variable denotes a rate of change relative to time. Since the rate of change of position \bm{\dot{\overrightarrow{r}}} is velocity \bm{\overrightarrow{v}} then \bm{\dot{\overrightarrow{r}}}=\bm{\overrightarrow{v}}.
- Similarly, two dots denotes taking the time derivative twice. \bm{\ddot{\overrightarrow{r}}}=\bm{\dot{\overrightarrow{v}}}=\bm{\overrightarrow{a}}.
- A hat denotes a unit vector, \bm{\hat{i}}, which has a magnitude of one. Unit vectors are useful for scaling a vector in a given direction.
General Kinematics
Defining the trajectory of an object is the main goal of kinematics. For the majority of engineering problems defining the position, velocity, and acceleration of an object is sufficient.
Position
\displaystyle\bm{\overrightarrow{r}=\int_{t_0}}^t\bm{\overrightarrow{v}}dt+\bm{\overrightarrow{r}}_0
Velocity
\displaystyle\bm{\overrightarrow{v}}=\dfrac{d\bm{\overrightarrow{r}}}{dt}=\dot{\bm{\overrightarrow{r}}}
\displaystyle\bm{\overrightarrow{v}}=\int_{t_0}^t\bm{\overrightarrow{a}}dt+\bm{\overrightarrow{v}}_0
Acceleration
\displaystyle\bm{\overrightarrow{a}}=\dfrac{d\bm{\overrightarrow{v}}}{dt}=\dot{\bm{\overrightarrow{v}}}=\dfrac{d^2\bm{\overrightarrow{r}}}{dt^2}=\ddot{\bm{\overrightarrow{r}}}
Rectangular Kinematics
\bm{\overrightarrow{r}}=x\bm{\hat{i}}+y\bm{\hat{j}}
\bm{\overrightarrow{v}}=\dot{x}\bm{\hat{i}}+\dot{y}\bm{\hat{j}}
\bm{\overrightarrow{a}}=\ddot{x}\bm{\hat{i}}+\ddot{y}\bm{\hat{j}}
Normal and Tangential Kinematics
ds=\rho d\beta
v=\dfrac{ds}{dt}=\rho\dot{\beta}
\bm{\overrightarrow{v}}=\rho\dot{\beta}\bm{\hat{e}}_t
\bm{\overrightarrow{a}}=\dot{v}\hat{e}_t+\dfrac{v^2}{\rho}\bm{\hat{e}}_n
a_t=\dot{v}
a_n=\dfrac{v^2}{\rho}
a=\sqrt{a_t^2+a_n^2}
\displaystyle\bm{\overrightarrow{a}}=\dfrac{d\bm{\overrightarrow{v}}}{dt}
\displaystyle\bm{\overrightarrow{a}}=\dfrac{d}{dt}(v\bm{\hat{e}}_t)
\displaystyle\bm{\overrightarrow{a}}=\dfrac{dv}{dt}\bm{\hat{e}}_t+v\dfrac{d\bm{\hat{e}}_t}{dt}
\displaystyle\dfrac{d\bm{\hat{e}}_t}{dt}=\dot{\beta}\bm{\hat{e}}_n=\dfrac{v}{\rho}\bm{\hat{e}}_n
\displaystyle\bm{\overrightarrow{a}}=\dot{v}\bm{\hat{e}}_t+\dfrac{v^2}{\rho}\bm{\hat{e}}_n
Polar Kinematics
\bm{\overrightarrow{r}}=r\bm{\hat{e}}_r
\bm{\overrightarrow{v}}=\dot{r}\bm{\hat{e}}_r+r\dot{\theta}\bm{\hat{e}}_\theta
v_r=\dot{r}
v_\theta=r\dot{\theta}
v=\sqrt{v_r^2+v_\theta^2}
\bm{\overrightarrow{a}}=(\ddot{r}-r\dot{\theta}^2)\bm{\hat{e}}_r+(r\ddot{\theta}+2\dot{r}\dot{\theta})\bm{\hat{e}}_\theta
a_r=\ddot{r}-r\dot{\theta}^2
a_\theta=r\ddot{\theta}+2\dot{r}\dot{\theta}
a=\sqrt{a_r^2+a_\theta^2}
\displaystyle\bm{\overrightarrow{v}}=\dfrac{d}{dt}(r\bm{\hat{e}}_r)
\displaystyle\bm{\overrightarrow{v}}=\dfrac{dr}{dt}\bm{\hat{e}}_r+r\dfrac{d\bm{\hat{e}}_r}{dt}
\displaystyle\dfrac{d\bm{\hat{e}}_r}{dt}=\dot{\theta}\bm{\hat{e}}_\theta
\displaystyle\bm{\overrightarrow{v}}=\dot{r}\bm{\hat{e}}_r+r\dot{\theta}\bm{\hat{e}}_\theta
\displaystyle\bm{\overrightarrow{a}}=\dfrac{d}{dt}(\dot{r}\bm{\hat{e}}_r+r\dot{\theta}\bm{\hat{e}}_\theta)
\displaystyle\bm{\overrightarrow{a}}=\dfrac{d\dot{r}}{dt}\bm{\hat{e}}_r+\dot{r}\dfrac{d\bm{\hat{e}}_r}{dt}+\dfrac{dr}{dt}\dot{\theta}\bm{\hat{e}}_\theta+r\dfrac{d\dot{\theta}}{dt}\bm{\hat{e}}_\theta+r\dot{\theta}\dfrac{d\bm{\hat{e}}_\theta}{dt}
\dfrac{d\bm{\hat{e}}_\theta}{dt}=-\dot{\theta}\bm{\hat{e}}_r
\bm{\overrightarrow{a}}=\ddot{r}\bm{\hat{e}}_r+\dot{r}\dot{\theta}\bm{\hat{e}}_\theta+\dot{r}\dot{\theta}\bm{\hat{e}}_\theta+r\ddot{\theta}\bm{\hat{e}}_\theta-r\dot{\theta}^2\bm{\hat{e}}_r
\bm{\overrightarrow{a}}=(\ddot{r}-r\dot{\theta}^2)\bm{\hat{e}}_r+(r\ddot{\theta}+2\dot{r}\dot{\theta})\bm{\hat{e}}_\theta
Relative Motion
Relative motion describes the motion of one object relative to another. The frame of reference may be stationary or moving. In reality all frames are relative (Albert Einstein).
For a static frame we may say that the (position, velocity, acceleration) of A relative to B is:
\bm{\overrightarrow{r}}_{AB}=\overrightarrow{r}_A-\overrightarrow{r}_B
\bm{\overrightarrow{v}}_{AB}=\overrightarrow{v}_A-\overrightarrow{v}_B
\bm{\overrightarrow{a}}_{AB}=\overrightarrow{a}_A-\overrightarrow{a}_B
Likewise, for a moving frame, C, we can define relative motion of A relative to C using the relative motion of A relative to B and the relative motion of B relative to C:
\bm{\overrightarrow{r}}_{AC}=\bm{\overrightarrow{r}}_{AB}+\bm{\overrightarrow{r}}_{BC}
\bm{\overrightarrow{v}}_{AC}=\bm{\overrightarrow{v}}_{AB}+\bm{\overrightarrow{v}}_{BC}
\bm{\overrightarrow{a}}_{AC}=\bm{\overrightarrow{a}}_{AB}+\bm{\overrightarrow{a}}_{BC}
Constrained Motion
Sometimes the motion of particles are constrained and can only act in a certain way. For example, particles connected within a rope will be constrained, adjacent rope particles will remain adjacent. We may describe how constrained a system is in terms of degrees of freedom. The number of degrees of freedom describes the number of independent variables that define a complete state of the system.
Circular Motion
For the special case of a particle following a circular path we can define the following parameters:
v_r=0
v_\theta=r\dot{\theta}
a_r=-r\dot{\theta}^2
a_\theta=r\ddot{\theta}
We can define the angular velocity in rad/s as \omega=\dot{\theta}. We can also define the angular acceleration in rad/s^2 as \alpha=\dot{\omega}.
v_r=0
v_\theta=r\omega
a_r=-r\omega^2
a_\theta=r\alpha