kinematics.md

jupyter
jupytext kernelspec text_representation extension format_name format_version jupytext_version .md markdown 1.2 1.6.0 display_name language name Python 3 python python3

Geometry of motion - kinematics

Position

Classical physics describes the position of an object using three independent coordinates e.g.

$$ \mathbf{r}_{P/O} = x\hat{i} + y\hat{j} + z\hat{k} $$ (position)

where $\mathbf{r}_{P/O}$ is the position of point $P$ with respect to the point of origin $O$, $x,~y,z$ are magnitudes of distance along a Cartesian coordinate system and $\hat{i},\hat{j}$ and $\hat{k}$ are unit vectors that describe three

import numpy as np
import matplotlib.pyplot as plt
plt.style.use('fivethirtyeight')

Velocity

The velocity of an object is the change in position, equation {eq}position, per length of time.

$$ \mathbf{v}{P/O} = \frac{d\mathbf{r}{P/O}}{dt} = \dot{x}\hat{i} + \dot{y}\hat{j} + \dot{z}\hat{k} $$ (velocity)

The notation $\dot{x}$ and $\ddot{x}$ is short-hand for writing out
$\frac{dx}{dt}$ and $\frac{d^2x}{dt^2}$, respectively.

The definition of velocity in equation {eq}velocity depends upon the change in position of all three independent coordinates, where $\frac{d}{dt}(x\hat{i})=\dot{x}\hat{i}$.

Remember the chain rule: $\frac{d}{dt}(x\hat{i})=\dot{x}\hat{i} +
x\dot{\hat{i}}$, but $\dot{\hat{i}}=0$ because this unit vector is not
changing direction. You'll see other unit vectors later that _do
change_.

{numref}postion-velocity

Example - GPS vs speedometer

You can find velocity based upon postion, but you can only find changes in position with velocity. Consider tracking the motion of a car driving down a road using GPS. You determine its motion and create the position, $\mathbf{r} = x\hat{i} +y\hat{j}$, where

$x(t) = 4t +3$ and $y(t) = 3t - 1$

To get the velocity, calculate $\mathbf{v} = \dot{\mathbf{r}}$

$\mathbf{v} = 4\hat{i} +3 \hat{j}$

t = np.arange(0,5)
x = 4*t + 3
y = 3*t -1
plt.plot(x,y,'o')
plt.quiver(x,y,4,3)
plt.title('Position of car on road every 1 second'+
    '\nvelocity shown as arrow')
plt.xlabel('x-position (m)')
plt.ylabel('y-position (m)');

Speed

The speed of an object is the magnitude of the velocity,

$|\mathbf{v}_{P/O}| = \sqrt{\mathbf{v}\cdot\mathbf{v}} = \sqrt{\dot{x}^2 + \dot{y}^2 + \dot{z}^2}$

Acceleration

The acceleration of an object is the change in velocity per length of time.

$\mathbf{a}{P/O} = \frac{d \mathbf{v}{P/O} }{dt} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k}$

where $\ddot{x}=\frac{d^2 x}{dt^2}$ and $\mathbf{a}_{P/O}$ is the acceleration of point $P$ with respect to the point of origin $O$.

Rotation and Orientation

The definitions of position, velocity, and acceleration all describe a single point, but dynamic engineering systems are composed of rigid bodies is needed to describe the position of an object.

from IPython.core.display import SVG

SVG(filename='./images/position_angle.svg')

In the figure above, the center of the block is located at $r{P/O}=x\hat{i}+y\hat{j}$ in both the left and right images, but the two locations are not the same. The orientation of the block is important for determining the position of all the material points._

In general, a rigid body has a pitch, yaw, and roll that describes its rotational orientation, as seen in the animation below. We will revisit 3D motion in Module_05

from IPython.display import YouTubeVideo
vid = YouTubeVideo("li7t--8UZms?loop=1")
display(vid)

Rotation in planar motion

Our initial focus is planar rotations e.g. yaw and roll are fixed. For a body constrained to planar motion, you need 3 independent measurements to describe its position e.g. $x$, $y$, and $\theta$