Which expression correctly gives the normal force on an incline in terms of mass m, gravity g, and angle theta?

Which expression correctly gives the normal force on an incline in terms of mass m, gravity g, and angle theta?

• A. F_N = m g sin theta
• B. F_N = m g cos theta
• C. F_N = m g
• D. F_N = m g tan theta

Answer: (B)

When a block sits on an incline, gravity acts straight downward, and the surface pushes perpendicular to itself. To see what the normal force does, split the weight into two components relative to the plane: a component pressing into the surface, mg cos theta, and a component pulling down along the slope, mg sin theta. Since there’s no acceleration into or away from the plane, the normal force must balance the perpendicular component of gravity. That makes the normal force F_N equal to m g cos theta. This also aligns with intuition: on a flat surface (theta = 0), F_N = m g; on a very steep incline, the perpendicular component gets smaller as theta increases toward 90 degrees. The other expressions aren’t the normal force because they correspond to the parallel component (mg sin theta), the total weight (mg), or a non-physical combination (mg tan theta) for this situation.

When a block sits on an incline, gravity acts straight downward, and the surface pushes perpendicular to itself. To see what the normal force does, split the weight into two components relative to the plane: a component pressing into the surface, mg cos theta, and a component pulling down along the slope, mg sin theta. Since there’s no acceleration into or away from the plane, the normal force must balance the perpendicular component of gravity. That makes the normal force F_N equal to m g cos theta. This also aligns with intuition: on a flat surface (theta = 0), F_N = m g; on a very steep incline, the perpendicular component gets smaller as theta increases toward 90 degrees. The other expressions aren’t the normal force because they correspond to the parallel component (mg sin theta), the total weight (mg), or a non-physical combination (mg tan theta) for this situation.