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A block of mass m rests on a rough horizontal surface. A horizontal force P acts on it along with another force Q directed at an angle theta to the vertical. For the block to remain in equilibrium, the minimum coefficient of friction between the block and the surface is:

1. (P + Q*sin(theta)) / (m*g + Q*cos(theta))
2. (P*cos(theta) + Q) / (m*g - Q*sin(theta))
3. (P + Q*cos(theta)) / (m*g + Q*sin(theta))
4. (P*sin(theta) + Q) / (m*g - Q*cos(theta))

Correct answer: (P + Q*sin(theta)) / (m*g + Q*cos(theta))

Solution

The force Q makes angle theta with the vertical, so its horizontal component is Q*sin(theta) and vertical component is Q*cos(theta). If Q is directed downward and to the side, the normal force N = m*g + Q*cos(theta). The net horizontal force requiring friction = P + Q*sin(theta). For equilibrium: mu >= (P + Q*sin(theta))/(m*g + Q*cos(theta)).

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