An object rests on an inclined plane. As the angle increases, the force needed to prevent the object from sliding down the incline increases.

• A. decreases
• B. increases
• C. remains the same
• D. none of these

Answer: (B)

As the incline gets steeper, the component of gravity pulling the block down the plane grows, while the friction available to resist that pull tends to shrink because the normal force N = mg cos theta decreases. That means the block is more likely to start sliding, and if you must supply an external force up the plane to keep it at rest, that force must increase with angle.

Quantitatively, the minimum external force needed to prevent sliding (when friction is at its maximum) is F_min = mg sin theta − mu_s mg cos theta, which is positive only after tan theta exceeds mu_s. The rate of change of that required force with angle is dF_min/d theta = mg(cos theta + mu_s sin theta), which is positive for angles between 0 and 90 degrees. So the necessary force grows as the incline becomes steeper. Of course, at very small angles friction alone can hold the block without any external force, but once you need any opposing force, it increases with angle.