Section 2.3: Forces on Inclined Surfaces
Objects on inclined planes experience components of weight along and perpendicular to the slope, along with normal and friction forces.
Weight Components:
\[
W_\parallel = mg \sin\theta, \quad W_\perp = mg \cos\theta
\]
Normal Force: Perpendicular contact force from the plane:
\[
N = W_\perp = mg \cos\theta
\]
Friction: Acts opposite motion along the slope:
\[
f_k = \mu_k N = \mu_k mg \cos\theta, \quad f_s \le \mu_s N
\]
Net Force Along Incline:
\[
F_{\text{net}} = W_\parallel - f
\]
Example 1
A 10 kg block slides down a frictionless incline of 30°. Find its acceleration along the slope.
\( a = g \sin\theta = 9.8 \sin30° = 4.9 \text{ m/s²} \)
Example 2
A 5 kg block on a 25° incline has μk = 0.2. Find friction and acceleration if moving down.
Normal force: \( N = mg \cos\theta = 5*9.8*\cos25° \approx 44.5 \text{ N} \)
Friction: \( f_k = μ_k N = 0.2*44.5 \approx 8.9 \text{ N} \)
Parallel weight: \( W_\parallel = mg\sin\theta = 5*9.8*\sin25° \approx 20.7 \text{ N} \)
Net force: \( F_{\text{net}} = 20.7 - 8.9 \approx 11.8 \text{ N} \)
Acceleration: \( a = F_{\text{net}}/m = 11.8/5 \approx 2.36 \text{ m/s²} \)
Practice Problems
- A 15 kg crate slides down a 30° frictionless incline. Find acceleration.
- For μk = 0.25 and m = 10 kg on a 20° incline, compute acceleration down the slope.
- A 12 kg block at rest on a 35° incline. Determine minimum μs to prevent sliding.
- Compute net force and acceleration for a 5 kg block sliding down 40° incline with μk = 0.2.
- Draw free-body diagram for a block on an inclined plane with friction.