Forest UCM NLM

From New IAC Wiki
Jump to navigation Jump to search


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] c[/math]) nor are microscopically small ( [math]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

TF UCM CartCoordSys.png


Vector Notation convention:

Position:

[math]\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]\vec{v}[/math] = [math]\frac{d \vec{r}}{dt}[/math] = [math]\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]\frac{d \hat{i}}{dt} =0 =\frac{d \hat{j}}{dt} =\frac{d \hat{k}}{dt}[/math]
[math]\vec{v}[/math] = [math]\frac{d \vec{r}}{dt}[/math] = [math]\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]\vec{a}[/math] = [math]\frac{d \vec{v}}{dt}[/math] = [math]\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

TF UCM PolarCoordSys.png Vector Notation convention:

Position:

Because [math]\hat{r}[/math] points in a unique direction given by [math]\hat{r} = \frac{\vec{r}}{|r|}[/math] we can write the position vector as

[math]\vec{r} = r \hat{r}[/math]
[math]\vec{r} \ne r \hat{r} +\phi \hat{\phi} [/math]: [math]\phi[/math] does not have the units of length


The unit vectors ([math]\hat{r}[/math] and [math]\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]\vec{r} = r \cos(\phi) \hat{i} + r \sin(\phi)\hat{j}[/math]

The dependence of position with [math]\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]t_1[/math] the particle is at [math]\vec{r}(t_1)[/math] and at time [math]t_2[/math] the particle is at [math]\vec{r}(t_2)[/math]


TF UCM PolarVectDiff.png


If we take the limit [math]t_2 \rightarrow t_1[/math] ( or [math]\Delta t \rightarrow 0[/math]) then we can write the velocity of this particle traveling in a circle as

[math]\hat{r} (t_2)-\hat{r}(t_1) \equiv \Delta \hat{r} = \Delta \phi \hat{\phi}[/math]
or
[math]\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]\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]\vec{v} = r \frac{d \phi}{dt} \hat{\phi}[/math]


If the particle is not constrained to circular motion ( i.e.: [math]r[/math] can change with time) then the velocity vector in polar coordinates is



[math]\vec{v}[/math] = [math]\frac{d r}{dt}\hat{r} + r\frac{d \phi}{dt} \hat{\phi}[/math]
or in more compact form
[math]\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]\equiv v_r [/math] Angular velocity [math]\equiv v_{\phi} [/math]

Acceleration in Polar Coordinates

Taking the derivative of velocity with time gives the acceleration


[math]\vec{a} = \frac{d \vec{v}}{dt} =\vec{\ddot{r}} [/math]
[math]= \frac{ d \left (\dot{r} \hat{r} + r \dot{\phi} \hat{\phi}= v_r \hat{r} + v_{\phi} \hat{\phi}\right)}{dt}[/math]
[math]= \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]= \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]\hat{\phi}[/math] with time.

Consider the position change below in terms of only the unit vector [math]\hat{\phi}[/math]


TF UCM PolarPhiUnitVectDiff.png


Using the same arguments used to calculate the rate of change in [math]\hat{r}[/math]:

If we take the limit [math]t_2 \rightarrow t_1[/math] ( or [math]\Delta t \rightarrow 0[/math]) then we can write the velocity of this particle traveling in a circle as

[math]\hat{\phi} (t_2)-\hat{\phi}(t_1) \equiv \Delta \hat{\phi} = \Delta \phi (- \hat{r})[/math]
or
[math]\frac{ d \hat{\phi}}{dt} = -\frac{d \phi}{dt} \hat{r}[/math]
[math]\frac{d \hat{\phi}}{dt}= -\dot{\phi} \hat{r}[/math]

Substuting the above into our calculation for acceleration:


[math]\vec{a} = \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]
[math]= \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} \left( -\dot{\phi} \hat{r}\right) \right )[/math]
[math]= \left ( \ddot{r} r \dot{\phi} -\dot{\phi}^2 \right) \hat{r} + \left ( 2\dot{r} \dot{\phi} +r \ddot{\phi} \right ) \hat{\phi} [/math]

Cylindrical

TF UCM CylCoordSys.png

Spherical

TF UCM SphericalCoordSys.png


Vectors

Scaler ( Dot ) product

Vector ( Cross ) product

Forest_Ugrad_ClassicalMechanics