Conservative Forces and Conservation of Energy (Proof)

Knowledge Roundtable > Tutorials > Conservative Forces…
Get Tutoring Info Varsity Tutors

Physics Tutorial

Conservative Forces and Conservation of Energy (Proof)

Intro

Search Tutors Varsity Tutors

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.

Search Tutors Varsity Tutors

Sample Problem

Search Tutors Varsity Tutors

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*}

Search Tutors Varsity Tutors

Solution

Search Tutors Varsity Tutors

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*}

Search Tutors Varsity Tutors
Get Tutoring Info Varsity Tutors

About The Author
Math And Physics From Algebra To College Calculus
I have been tutoring math and physics on and off for 10 years now. I have been involved in various residential tutoring programs as well as college level teaching. I can tutor mathematics from middle school algebra all the way to college level advanced calculus. My standard teaching method is...
8 Subjects
KnowRo Tutor
2 Tutorials
Amarillo, TX
Get Tutoring Info

Suggested Tutors for Physics Help

Lindsey C

East Kingston, NH

Spanish Tutor, English Tutor, Writing Professional

Brent A

Seabrook, NH

Chemistry And Physics

Altai P

Durham, NH

Science And Math Tutor

Paul P

Dover, NH

Math Subject Matter Expert And SAT Instructor

Nearby tutors offer these subjects:

Science Tutors in Exeter NH, Biology Tutors in Exeter NH, Chemistry Tutors in Exeter NH, Physics Tutors in Exeter NH, Astronomy Tutors in Exeter NH, Physical Science Tutors in Exeter NH

Tutors in nearby towns:

Tutors in TX, Tutors in Amarillo TX
^