Revision as of 20:20, 19 June 2014

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.

Forest_UCM_NLM_Ch1_CoordSys

Vectors

Scaler ( Dot ) product

Vector ( Cross ) product

Forest_Ugrad_ClassicalMechanics

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 [[Forest_UCM_NLM_Ch1_CoordSys]]
-===Cartesian===
-[[File:TF_UCM_CartCoordSys.png| 200 px]]
-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===
-[[File:TF_UCM_PolarCoordSys.png| 200 px]]
-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>
-[[File:TF_UCM_PolarVectDiff.png| 200 px]]
-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>
-[[File:TF_UCM_PolarPhiUnitVectDiff.png| 400 px]]
-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}^2 \right) \hat{r}  + \left ( 2\dot{r} \dot{\phi} +r \ddot{\phi} \right ) \hat{\phi} </math>
-For the case of circular motion at constant <math> r (\dot{r} = 0)</math>
-:<math>\vec{a} =   -r\dot{\phi}^2 \hat{r}  + r \ddot{\phi} \hat{\phi} </math>
-radial (centripetal, center seeking)  acceleration <math>\equiv   -r\dot{\phi}^2 \hat{r} = -r \omega^2 \hat{r}</math>
-angular (tangential)  acceleration <math>\equiv   r \ddot{\phi} \hat{\phi} = r \alpha \hat{\phi}</math>
-If <math>\dot{r} \ne 0</math>
-Then there are two additional terms
-:<math>\ddot{r} \hat {r}</math> = radial acceleration
-:<math>2\dot{r} \dot{\phi}  \hat {\phi}</math> = Coriolis acceleration (to be described later)
-===Cylindrical===
-[[File:TF_UCM_CylCoordSys.png| 200 px]]
-===Spherical===
-[[File:TF_UCM_SphericalCoordSys.png| 200 px]]
 ==Vectors==

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