Forest UCM EnergyIntPart

Energy of Interacting particles

Translational invariance

Consider two particles that interact via a conservative force [math]\vec{F}[/math]

Let [math]\vec{r}_1[/math] identify the location of object 1 from an arbitrary reference point and [math]\vec{r}_2[/math] locate the second object.

The vector that points from object 2 to object 1 may be written as

[math]\vec r = \vec{r}_1 - \vec{r}_2[/math]

The distance between the two object is given as the length of the above vector

If the force is a central force

[math]\vec F = \frac{k}{r^3} \vec r = \frac{k}{\left | r_1 - r_2 \right |^3} \left ( \vec{r}_1 - \vec{r}_2 \right )[/math]

Notice

The interparticle force is independent of the coordinate system's position, only the difference betweenthe positions matters

If object 2 was fixed so it is not accelerating and we place the origin of the coordinate system on object 2

Then the force is that of a single object

One potential for Both Particles

Both forces from same potential

If the above force is conservative then a potential exists such that

[math]\vec{F}_{12} = - \vec{\nabla}_1 U(\vec{r}_1) [/math]

[math] = - \vec{\nabla}_1 U(\vec{r}_1) = - \left( \hat i \frac{\partial}{\partial x_1} + \hat j \frac{\partial}{\partial y_1} + \hat k \frac{\partial}{\partial z_1} \right ) U(\vec{r}_1)[/math]

[math]= - \vec{\nabla}_1 U(\vec{r}_1 - \vec{r}_2) [/math]

Newton's 3rd law requries that

[math]\vec{F}_{21} = - \vec{F}_{12} = - \left (- \vec{\nabla}_1 U(\vec{r}_1) \right ) = \vec{\nabla}_1 U(\vec{r}_1 - \vec{r}_2) [/math]

[math]- \left (- \vec{\nabla}_2 U(\vec{r}_2) \right ) = -\vec{\nabla}_2 U(\vec{r}_1 - \vec{r}_2)[/math]

or

[math] \vec{\nabla}_1 U(\vec{r}_1 - \vec{r}_2) = -\vec{\nabla}_2 U(\vec{r}_1 - \vec{r}_2)[/math]

You can find the net external force on a body in the system once you have the potential for the system

Elastic Collisions

Definition

BOTH Momentum and Energy are conserved in an elastic collision

Example

Consider two object that collide elastically

Conservation of Momentum

[math]\left ( p_1 + p_2 \right ) _{\mbox{initial}} = \left ( p_1 + p_2 \right ) _{\mbox{final}}[/math]

Conservation of Energy

[math]\left ( T + U \right ) _{\mbox{initial}} = \left ( T + U \right ) _{\mbox{final}}[/math]

When the initial and final states are far away fromthe collision point

[math]U_{\mbox{initial}} = U_{\mbox{final}} = 0 =[/math] arbitrary constant

Example

Consider an elastic collision between two equal mass objecs one of which is at rest.

Conservation of momentum

[math] m \vec{v}_1 = m \left (\vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime} \right )[/math]

Conservation of Energy

[math] \frac{1}{2} m v_1^2 = \frac{1}{2} m \left (v_1^{\;\prime} \right )^2 + \frac{1}{2} m\left ( v_2^{\;\prime} \right )^2[/math]

Square the conservation of momentum equation

[math] \vec{v}_1 \cdot \vec{v}_1 = \left (\vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime} \right ) \cdot \left (\vec{v}_1^{\;\prime} + \vec{v}_2^{\;\prime} \right )[/math]

[math] v_1^2 = \left (v_1^{\;\prime} \right )^2 + \left ( v_2^{\;\prime} \right )^2 + 2 \vec{v}_1^{\;\prime} \cdot \vec{v}_2^{\;\prime} [/math]

compare the above conservation of momentum equation with the conservation of energy equation

[math] v_1^2 = \left (v_1^{\;\prime} \right )^2 + \left ( v_2^{\;\prime} \right )^2[/math]

and you conclude that

[math]2 \vec{v}_1^{\;\prime} \cdot \vec{v}_2^{\;\prime} = 0 \;\;\;\; \Rightarrow \vec{v}_1^{\;\prime} \perp \vec{v}_2^{\;\prime} [/math]

Forest_UCM_Energy#Energy_of_Interacting_Particles