A 4-kg block on a 30° incline slides with kinetic friction μk = 0.3. Find its acceleration down the plane (g = 9.8 m/s^2).

A 4-kg block on a 30° incline slides with kinetic friction μk = 0.3. Find its acceleration down the plane (g = 9.8 m/s^2).

• A. N ≈ 33.9 N; F_par ≈ 19.6 N; F_f ≈ 10.17 N; F_net ≈ 9.43 N; a ≈ 2.36 m/s^2 down the plane.
• B. N ≈ 49 N; F_par ≈ 25 N; F_f ≈ 15 N; a ≈ 2 m/s^2
• C. N ≈ 20 N; F_par ≈ 5 N; F_f ≈ 1 N; a ≈ 0.5 m/s^2
• D. N ≈ 60 N; F_par ≈ 30 N; F_f ≈ 18 N; a ≈ 3 m/s^2 down the plane

Answer: (A)

The key idea is that motion on an incline is governed by the net force along the plane, which equals the downslope component of gravity minus the kinetic friction, and the acceleration is that net force divided by the mass. Calculate the normal force: N = m g cos θ = 4 × 9.8 × cos 30° ≈ 33.9 N. The component of gravity pulling down the plane: F_par = m g sin θ = 4 × 9.8 × sin 30° = 19.6 N. Friction opposes the motion with magnitude F_f = μk N = 0.3 × 33.9 ≈ 10.2 N. Net force along the plane: F_net = F_par − F_f ≈ 19.6 − 10.2 ≈ 9.4 N. Acceleration down the plane: a = F_net / m ≈ 9.4 / 4 ≈ 2.36 m/s². This matches the set of values where N ≈ 33.9 N, F_par ≈ 19.6 N, F_f ≈ 10.17 N, F_net ≈ 9.43 N, and a ≈ 2.36 m/s² down the plane. The normal force is mg cos θ, not mg, which is why it’s about 33.9 N rather than 49 N.

The key idea is that motion on an incline is governed by the net force along the plane, which equals the downslope component of gravity minus the kinetic friction, and the acceleration is that net force divided by the mass.

Calculate the normal force: N = m g cos θ = 4 × 9.8 × cos 30° ≈ 33.9 N.

The component of gravity pulling down the plane: F_par = m g sin θ = 4 × 9.8 × sin 30° = 19.6 N.

Friction opposes the motion with magnitude F_f = μk N = 0.3 × 33.9 ≈ 10.2 N.

Net force along the plane: F_net = F_par − F_f ≈ 19.6 − 10.2 ≈ 9.4 N.

Acceleration down the plane: a = F_net / m ≈ 9.4 / 4 ≈ 2.36 m/s².

This matches the set of values where N ≈ 33.9 N, F_par ≈ 19.6 N, F_f ≈ 10.17 N, F_net ≈ 9.43 N, and a ≈ 2.36 m/s² down the plane. The normal force is mg cos θ, not mg, which is why it’s about 33.9 N rather than 49 N.