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A hollow sphere of mass m rests on a plank of mass M lying on a smooth inclined plane of inclination angle theta. Friction between the sphere and the plank is sufficient to prevent slipping. Let mu be the minimum coefficient of friction between the plank and the incline required to keep the plank stationary. Which of the following statements are correct?

1. When the plank is at rest, the friction force between the sphere and the plank is 2mg sin(theta)/5
2. The minimum coefficient of friction is mu = (5M + 2m) tan(theta) / [5(M + m)]
3. When there is no friction between plank and incline, the acceleration of the plank is (2m + 5M)g sin(theta) / (5M)
4. When there is no friction between plank and incline, the friction force on the sphere is zero

Correct answer: When there is no friction between plank and incline, the friction force on the sphere is zero

Solution

When no friction acts between the plank and the incline, solving the coupled equations shows both the sphere and the plank accelerate at g sin(theta) down the incline, so there is no relative motion or tendency of slipping between them, making the contact friction on the sphere zero. Option A is also correct (friction = 2mg sin(theta)/5 when plank is stationary), and option B gives the correct mu. Option C is incorrect because the plank accelerates at g sin(theta), not the formula stated.

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