Discussion
A particle of mass \(m\) moves in a potential given by
\[ V(x,y,z) = -k\frac{y}{(x^{2}+y^{2}+z^{2})}, \]
where \(k\) is a constant.
Which of the following statements is correct?
Answer: B
The correct statement is that angular momentum of the system is not conserved.
The potential \(V = -k\,y/r^2\) depends on direction as well as on \(r = \sqrt{x^2 + y^2 + z^2}\). A central force needs a potential that depends on \(r\) alone; this one does not, so the force has components across the radius and exerts a torque \(\vec{r} \times \vec{F} \neq 0\). Angular momentum therefore changes.
The force is not central, so that statement is false. \(V\) depends on \(y\), so \(F_y = -\partial V/\partial y \neq 0\) and linear momentum along \(y\) is not conserved. \(V\) does not depend on time, so the force is conservative and the total energy is conserved, which rules out the last statement.
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