Forest UCM NLM
Newton's Laws of Motion
Limits of Classical Mechanic
Classical Mechanics is the formulations of physics developed by Newton (1642-1727), Lagrange(1736-1813), and Hamilton(1805-1865).
It may be used to describe the motion of objects which are not moving at high speeds (0.1[math]\displaystyle{ c }[/math]) nor are microscopically small ( [math]\displaystyle{ 10^{-9} m }[/math]).
The laws are formulated in terms of space, time, mass, and force:
Space and Time
Space
Cartesian, Spherical, and Cylindrical coordinate systems are commonly used to describe three-dimensional space.
Cartesian
Vector Notation convention:
Position:
- [math]\displaystyle{ \vec{r} = x \hat{i} + y \hat{j} + z \hat{k} = (x,y,z) = \sum_1^3 r_i \hat{e}_i }[/math]
Velocity:
- [math]\displaystyle{ \vec{v} }[/math] = [math]\displaystyle{ \frac{d \vec{r}}{dt} }[/math] = [math]\displaystyle{ \frac{d x}{dt}\hat{i} + x\frac{d \hat{i}}{dt} + \cdots }[/math]
cartesian unit vectors do not change with time (unit vectors for other coordinate system types do)
- [math]\displaystyle{ \frac{d \hat{i}}{dt} =0 =\frac{d \hat{j}}{dt} =\frac{d \hat{k}}{dt} }[/math]
- [math]\displaystyle{ \vec{v} }[/math] = [math]\displaystyle{ \frac{d \vec{r}}{dt} }[/math] = [math]\displaystyle{ \frac{d x}{dt}\hat{i} + \frac{d y}{dt}\hat{j} + \frac{d z}{dt}\hat{k} }[/math]
Similarly Acceleration is given by
- [math]\displaystyle{ \vec{a} }[/math] = [math]\displaystyle{ \frac{d \vec{v}}{dt} }[/math] = [math]\displaystyle{ \frac{d^2 x}{dt^2}\hat{i} + \frac{d^2 y}{dt^2}\hat{j} + \frac{d^2 z}{dt^2}\hat{k} }[/math]
Polar
Vector Notation convention:
Position:
Because [math]\displaystyle{ \hat{r} }[/math] points in a unique direction given by [math]\displaystyle{ \hat{r} = \frac{\vec{r}}{|r|} }[/math] we can write the position vector as
- [math]\displaystyle{ \vec{r} = r \hat{r} }[/math]
- [math]\displaystyle{ \vec{r} \ne r \hat{r} +\phi \hat{\phi} }[/math]: [math]\displaystyle{ \phi }[/math] does not have the units of length
The unit vectors ([math]\displaystyle{ \hat{r} }[/math] and [math]\displaystyle{ \hat{\phi} }[/math] ) are changing in time. You could express the position vector in terms of the cartesian unit vectors in order to avoid this
- [math]\displaystyle{ \vec{r} = r \cos(\phi) \hat{i} + r \sin(\phi)\hat{j} }[/math]
The dependence of position with [math]\displaystyle{ \phi }[/math] can be seen if you look at how the position changes with time.
Velocity in Polar Coordinates
Consider the motion of a particle in a circle. At time [math]\displaystyle{ t_1 }[/math] the particle is at [math]\displaystyle{ \vec{r}(t_1) }[/math] and at time [math]\displaystyle{ t_2 }[/math] the particle is at [math]\displaystyle{ \vec{r}(t_2) }[/math]
If we take the limit [math]\displaystyle{ t_2 \rightarrow t_1 }[/math] ( or [math]\displaystyle{ \Delta t \rightarrow 0 }[/math]) then we can write the velocity of this particle traveling in a circle as
- [math]\displaystyle{ \hat{r} (t_2)-\hat{r}(t_1) \equiv \Delta \hat{r} = \Delta \phi \hat{\phi} }[/math]
- or
- [math]\displaystyle{ \frac{ d \hat{r}}{dt} = \frac{d \phi}{dt} \hat{\phi} }[/math]
Thus for circular motion at a constraint radius we get the familiar expression
- [math]\displaystyle{ \vec{v} = \lim_{\Delta t \rightarrow 0} \frac{\vec{r}(t_2)-\vec{r}(t_1)}{\Delta t}= \lim_{\Delta t \rightarrow 0} \frac{r\left( \hat{r}(t_2) - \hat{r}(t_1)\right)}{\Delta t} = r \frac{\Delta \phi}{\Delta t} \hat{\phi} = r \omega \hat{\phi} }[/math]
- [math]\displaystyle{ \vec{v} = r \frac{d \phi}{dt} \hat{\phi} }[/math]
If the particle is not constrained to circular motion ( i.e.: [math]\displaystyle{ r }[/math] can change with time) then the velocity vector in polar coordinates is
- [math]\displaystyle{ \vec{v} }[/math] = [math]\displaystyle{ \frac{d r}{dt}\hat{r} + r\frac{d \phi}{dt} \hat{\phi} }[/math]
- or in more compact form
- [math]\displaystyle{ \vec{v}=\vec{\dot{r}} = \dot{r} \hat{r} + r \dot{\phi} \hat{\phi}= v_r \hat{r} + v_{\phi} \hat{\phi} }[/math]
- linear velocity [math]\displaystyle{ \equiv v_r }[/math] Angular velocity [math]\displaystyle{ \equiv v_{\phi} }[/math]
Acceleration in Polar Coordinates
Taking the derivative of velocity with time gives the acceleration
- [math]\displaystyle{ \vec{a} = \frac{d \vec{v}}{dt} =\vec{\ddot{r}} }[/math]
- [math]\displaystyle{ = \frac{ d \left (\dot{r} \hat{r} + r \dot{\phi} \hat{\phi}= v_r \hat{r} + v_{\phi} \hat{\phi}\right)}{dt} }[/math]
- [math]\displaystyle{ = \left ( \frac{ d \dot{r}}{dt} \hat{r} + \dot{r} \frac{ d\hat{r}}{dt} \right) + \left ( \frac{d r}{dt} \dot{\phi} \hat{\phi} +r \frac{d \dot{\phi}}{dt} \hat{\phi} +r \dot{\phi} \frac{d \hat{\phi}}{dt} \right ) }[/math]
- [math]\displaystyle{ = \left ( \ddot{r} \hat{r} + \dot{r} \dot{\phi}\hat{\phi} \right) + \left ( \dot{r} \dot{\phi} \hat{\phi} +r \ddot{\phi} \hat{\phi} +r \dot{\phi} \frac{d \hat{\phi}}{dt} \right ) }[/math]
We need to find the derivative of the unit vector [math]\displaystyle{ \hat{\phi} }[/math] with time.
Consider the position change below in terms of only the unit vector [math]\displaystyle{ \hat{\phi} }[/math]
- [math]\displaystyle{ \frac{d \hat{\phi}}{dt}= -\dot{\phi} \hat{r} }[/math]
Cylindrical
Spherical
