The image describes a physical scenario where a mass $m$ is attached to the end of a massless pole and connected to a platform with mass $M$. The setup involves a frictionless table, and the question asks for the minimum static friction coefficient $\mu$ between the platform and the table so that the platform remains stationary while the mass $m$ rotates around the pole.

Key elements to extract:

1. System setup:

   • A mass $m$ is attached to a massless pole of length $l$ and is released from rest.
   • The platform has mass $M$, and the goal is to prevent it from moving.
   • The mass $m$ rotates around the pole in a circular motion.

2. Centripetal force:

   • When the mass $m$ moves in a circular path, the tension in the rope provides the centripetal force.
   • The formula for centripetal force is $F_{\text{centripetal}} = \frac{m v^2}{l}$, where $v$ is the tangential velocity of the mass $m$, and $l$ is the length of the rope.

3. Friction force on platform:

   • To prevent the platform from moving, the friction force between the platform (mass $M$) and the table must balance the horizontal force exerted by the tension in the rope.
   • The friction force $F_{\text{friction}} = \mu M g$, where $\mu$ is the static friction coefficient, and $g$ is gravitational acceleration.

4. Force equilibrium:

   • The tension in the rope causes a force on the platform, which is opposed by friction. Hence, $T = F_{\text{friction}}$.

Solution Approach:

• The tension in the rope can be expressed as $T = \frac{m v^2}{l}$.
• The platform remains stationary if the friction force equals this tension: $\mu M g = T$.
• Substituting $T$: $\mu M g = \frac{m v^2}{l}$
• Rearranging for $\mu$: $\mu = \frac{m v^2}{M g l}$

However, since no specific velocity is given, the expression in this form represents the minimum static friction coefficient $\mu$ that depends on how fast the mass $m$ is rotating.

Would you like to dive into more details on how we can compute the velocity $v$, or do you have any other specific questions about this setup?

Here are five follow-up questions to expand on this concept:

1. How would the coefficient of friction change if the mass $M$ of the platform increases?
2. What role does the length of the rope $l$ play in determining the tension?
3. Can the mass $m$ ever move fast enough that the platform moves, even with high friction?
4. What assumptions are we making about the rotational motion of the mass $m$?
5. How can the velocity $v$ of the mass be calculated in this scenario?

Tip:

Always analyze the forces acting on each component of the system independently and identify which force provides the centripetal acceleration for circular motion.