Answer
(a) $F = mg~tan~\theta$
(b) $F = mg~tan~\theta$
(c) $F = \frac{mg~sin~\theta + ma}{cos~\theta}$
Work Step by Step
(a) The component of the applied force directed up the ramp must be equal in magnitude to the component of the crate's weight directed down the ramp:
$F~cos~\theta = mg~sin~\theta$
$F = mg~tan~\theta$
(b) If the crate is moving at a constant speed, then the net force on the crate must be zero. The component of the applied force directed up the ramp must be equal in magnitude to the component of the crate's weight directed down the ramp:
$F~cos~\theta = mg~sin~\theta$
$F = mg~tan~\theta$
(c) $\sum F = ma$
$F~cos~\theta-mg~sin~\theta = ma$
$F~cos~\theta = mg~sin~\theta + ma$
$F = \frac{mg~sin~\theta + ma}{cos~\theta}$
