Question

A particle $P$ of mass $m$ is constrained to move on the surface of a cylinder under a force $-k\vec{r}$ as shown in the figure ($k$ is a positive constant). Neglect friction. Which of the following statements is correct?

• A. Total energy of the particle is not conserved.
• B. The motion along $z$ direction is simple harmonic.
• C. Angular momentum of the particle about $O$ increases with time.
• D. Linear momentum of the particle is conserved.

Hint:

When a restoring force is proportional to displacement along a direction, motion in that direction is always simple harmonic.

Answer 1

The Correct Option is B

Step 1: Understand the constraint.
The particle moves on the surface of a vertical cylinder → radius is fixed, say $r=R$. Its position can change in $\phi$ (around axis) and $z$ (along height).

Step 2: Examine the force $-k\vec{r$.}
$\vec{r}$ has components in the radial and $z$ directions. But the radial motion is constrained; thus only the $z$-component of force acts: $F_z = -k z.$

Step 3: Identify the type of motion.
Equation of motion in $z$: $\displaystyle m\ddot{z} = -k z.$ This is exactly the equation of simple harmonic motion with frequency $\displaystyle \omega=\sqrt{\frac{k}{m}}.$

Step 4: Check the other options.
• Total energy is conserved because the force is conservative. • Angular momentum about the axis is not increasing—there is no torque about $z$ axis. • Linear momentum is not conserved due to the central restoring force.

Step 5: Conclusion.
Only the motion along the $z$ direction is SHM, so option (B) is correct.