STEP II 2007 question 10 solution

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Let X make an angle to the horizontal plane and let Y be perpendicular to X. By symmetry,

The mass of the cylinder is . The mass of the hemisphere is

The centre of mass of the cylinder is (r, 1.5r). The centre of mass of the hemisphere is (r, 3.375r)

Resolving vertically: Resolving horizontally:

Before we take moments about the point of the cylinder in contact with the ground (let's call it point A), it is necessary to find the distances involved.

By referring to the diagram, note that the horizontal distance between A and the weight is .

Also, the vertical distance between A and P is .

Taking moments about A:

When this expression is negative, the force P acts in the opposite direction to that shown in the diagram. F must also act in the opposite direction.

The magnitude of P is

Solution by Dystopia.