Introduction
When an object sits on or moves along a slope (an inclined plane), several forces act on it. The key ideas are gravity pulling downward, the normal force from the surface, and friction that can resist motion. The component of gravity parallel to the slope causes sliding, while the component perpendicular to the slope is balanced by the normal force.
1) Visualizing the forces
Imagine a block on a ramp inclined at an angle θ from the horizontal.
- Gravity (weight) acts straight down toward the Earth: W = m g.
- Normal force is the force exerted by the ramp perpendicular to its surface, pushing the block away from the ramp.
- Friction depends on whether the block is at rest or sliding. Kinetic friction acts opposite the direction of motion; static friction exists up to a maximum value.
2) Resolving gravity into components
To analyze motion on the slope, resolve gravity into two components relative to the ramp: parallel to the slope and perpendicular to the slope.
- Parallel component: W∥ = m g sin(θ)
- Perpendicular component: W⊥ = m g cos(θ)
The normal force N balances W⊥: N = m g cos(θ).
3) Motion without friction
If there is no friction, the net force along the slope is Fnet,∥ = m g sin(θ). This leads to acceleration a = g sin(θ) down the slope.
4) Including friction
Friction opposes motion. Its maximum static friction is f_s,max = μs N. If the needed force to start moving (m g sin(θ)) is less than f_s,max, the block stays at rest. Once it moves, kinetic friction f_k = μk N opposes motion, giving a = g(sin(θ) - μk cos(θ)) down the slope (assuming sliding down).
- Static friction prevents motion until θ is large enough or applied force increases.
- For a given μk, if sin(θ) > μk cos(θ), the block accelerates downward.
5) Step-by-step example
- Block mass: m = 2.0 kg; incline angle: θ = 30°; μs = 0.4; μk = 0.3; g ≈ 9.8 m/s².
- Compute components: W∥ = m g sin(30°) = 2 × 9.8 × 0.5 ≈ 9.8 N; W⊥ = m g cos(30°) = 2 × 9.8 × (√3/2) ≈ 16.97 N.
- Normal force N = W⊥ ≈ 16.97 N.
- Maximum static friction f_s,max = μs N ≈ 0.4 × 16.97 ≈ 6.79 N.
- Required to start sliding: F_required = W∥ ≈ 9.8 N. Since 9.8 N > 6.79 N, the block will start to slide.
- After sliding, acceleration a = g (sinθ − μk cosθ) ≈ 9.8 (0.5 − 0.3 × 0.866) ≈ 9.8 (0.5 − 0.2598) ≈ 9.8 × 0.2402 ≈ 2.35 m/s² down the slope.
6) Key takeaways
- The gravity component parallel to the slope drives motion; the perpendicular component is balanced by the normal force.
- Friction depends on the normal force and the coefficient of friction; static friction prevents motion up to its limit, while kinetic friction acts during motion.
- For a given incline, if sin(θ) > μk cos(θ), the block will accelerate down the slope when sliding.
7) Quick practice questions
- What is the parallel component of gravity on a 45° incline for mass m?
- Given μs and N, how do you determine if an object starts to slide?
- How does increasing the incline angle affect the likelihood of sliding?