Solution Found!
Both the Coulomb and gravitational forces lead to
Chapter 4, Problem 4.49(choose chapter or problem)
Both the Coulomb and gravitational forces lead to potential energies of the form U = y r21, where y denotes ka1q2 in the case of the Coulomb force and Gm 1m2 for gravity, and r1 and r2 are the positions of the two particles. Show in detail that V1 U is the force on particle 1 and V2 U that on particle 2.
Questions & Answers
QUESTION:
Both the Coulomb and gravitational forces lead to potential energies of the form U = y r21, where y denotes ka1q2 in the case of the Coulomb force and Gm 1m2 for gravity, and r1 and r2 are the positions of the two particles. Show in detail that V1 U is the force on particle 1 and V2 U that on particle 2.
ANSWER:Step 1 of 3
Let’s define \( \mathbf{r} = \mathbf{r}_1-{\mathbf{r}}_2\), so that \(\mathbf{r}\) is a vector pointing from particle 2 to particle 1.
The \(PE\) is \(U = \gamma /r\), the force on particle 1 due to particle 2 is \(\mathbf{F}_{12} = -(\gamma/r^2)\hat{\mathbf{r}}\), and that on particle 2 due to particle 1 is
\(\mathbf{F}_{21} = -(\gamma/r^2)\hat{\mathbf{r}}\).
We'll start with the \(x\) component of \(-\triangledown _1U\):