Conservative Forces and Conservation of Energy (Proof)

Physics Tutorial

Conservative Forces and Conservation of Energy (Proof)

Intro

In this short section we will

a) define conservative forces and
b) prove that such forces conserve a quantity we call energy for constant mass systems.

A conservative force \vec{F} is defined as:

(1)   \begin{equation*} \vec{F} = - \nabla U(\vec{r}) \end{equation*}

Where U is a function of the coordinates.Now we multiply both sides by \vec{v} = d \vec{x} / dt. We get:

(2)   \begin{equation*} \vec{v}\cdot \vec{F} = m \vec{v} \cdot \frac{d \vec{v}}{dt} = - \vec{v} \cdot \nabla U. \end{equation*}

Where we used Newton’s second law in the first equal sign. It is left to the reader to prove:

(3)   \begin{equation*} \vec{v} \cdot \frac{d \vec{v}}{dt} = \frac{1}{2} \frac{d v^2}{dt} \,\,\, \text{and} \,\,\, \vec{v}\cdot \nabla U = \frac{d U}{dt} \end{equation*}

Where v is the velocity. Using the equations above we have:

(4)   \begin{equation*} \frac{d}{dt} \left( \frac{1}{2} m v^2 + U \right) = 0. \end{equation*}

Defining energy to be the sum of the kinetic energy m v^2/2 and the potential energy U we have:

(5)   \begin{equation*} \frac{dE}{dt} = 0. \end{equation*}

In other words E_{final} = E_{initial}, the standard conservation of energy.

Sample Problem

Prove

(1)   \begin{equation*} \vec{v} \cdot \frac{d \vec{v}}{dt} = \frac{1}{2} \frac{d v^2}{dt} \,\,\, \text{and} \,\,\, \vec{v}\cdot \nabla U = \frac{d U}{dt} \end{equation*}

Solution

For the first equality we have:

    \begin{align*} \frac{1}{2} \frac{d v^2}{dt} &= \frac{1}{2} \frac{ d( v_x^2 + v_y^2 +v_z^2)}{dt} \\ &= v_x \frac{dv_x}{dt} + v_y \frac{dv_y}{dt}+ v_z \frac{dv_z}{dt} \\ &= \vec{v} \cdot \frac{d \vec{v}}{dt}. \end{align*}

For the second equality we have:

(1)   \begin{equation*} \vec{v} \cdot U(x,y,z) = \frac{d x}{dt} \frac{\partial U}{\partial x} + \frac{d y}{dt} \frac{\partial U}{\partial y} + \frac{d z}{dt} \frac{\partial U}{\partial z} = \frac{d U }{dt}. \end{equation*}



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