Section 2.3: Forces on Inclined Surfaces

This section explains how forces act on objects placed on inclined planes. We analyze the components of weight, normal force, and friction, and how to calculate acceleration along the incline.

Weight Components:

For an object of mass \( m \) on an incline of angle \( \theta \):

  • Parallel to incline: \( F_{\parallel} = mg \sin\theta \)
  • Perpendicular to incline: \( F_{\perp} = mg \cos\theta \)

Normal force: \( F_N = F_{\perp} = mg \cos\theta \)

Friction on Incline:

Kinetic friction: \( f_k = \mu_k F_N = \mu_k mg \cos\theta \)

Static friction: \( f_s \leq \mu_s F_N \)

Net Force along Incline:

\( F_{\text{net}} = F_{\parallel} - f_k \)

Acceleration: \( a = \frac{F_{\text{net}}}{m} \)

Example 1

A 10 kg block is on a 30° incline. The coefficient of kinetic friction is 0.1. Find acceleration down the incline.

Weight component along incline: \( F_\parallel = mg \sin\theta = 10*9.8*\sin30° \approx 49 \text{ N} \)

Normal force: \( F_N = mg \cos\theta = 10*9.8*\cos30° \approx 84.9 \text{ N} \)

Friction: \( f_k = \mu_k F_N = 0.1*84.9 \approx 8.49 \text{ N} \)

Net force: \( F_\text{net} = 49 - 8.49 \approx 40.51 \text{ N} \)

Acceleration: \( a = F_\text{net}/m = 40.51/10 \approx 4.05 \text{ m/s²} \)

Example 2

A 5 kg block on a 45° incline has µk = 0.2. Find acceleration down the incline.

F_\parallel = 5*9.8*sin45° ≈ 34.65 N

F_N = 5*9.8*cos45° ≈ 34.65 N

f_k = 0.2*34.65 ≈ 6.93 N

F_net = 34.65 - 6.93 ≈ 27.72 N

a = F_net / m = 27.72 / 5 ≈ 5.54 m/s²

Practice Problems

  1. A 12 kg block on a 20° incline has µk = 0.15. Find acceleration.
  2. A 10 kg crate slides down a 30° incline with friction 10 N. Compute acceleration.
  3. A 15 kg box on a 25° incline has µk = 0.2. Find net force and acceleration.
  4. A 8 kg object on 40° incline, µk = 0.1. Determine acceleration.
  5. A 20 kg block slides down 35° incline, friction = 30 N. Compute acceleration.
  6. A 5 kg object on 30° incline, frictionless. Find acceleration.
  7. A 7 kg block on 45° incline, µk = 0.25. Find acceleration down incline.
  8. A 12 kg box on 15° incline, µk = 0.1. Compute net force along incline.
  9. A 10 kg block on 10° incline, µk = 0.05. Determine acceleration.
  10. A 6 kg block on 50° incline, µk = 0.2. Compute net force and acceleration.
  11. A 18 kg box slides down 30° incline with µk = 0.15. Find acceleration.
  12. A 25 kg block on 20° incline, friction = 40 N. Find acceleration.
  13. A 9 kg block slides on 35° incline, µk = 0.2. Compute net force.
  14. A 4 kg object on 45° incline, µk = 0.1. Find acceleration down incline.
  15. A 16 kg block slides on 25° incline, friction = 20 N. Compute acceleration.
  16. A 10 kg crate on 30° incline, µk = 0.2. Determine acceleration.
  17. A 7 kg block on 40° incline, friction = 8 N. Compute net force.
  18. A 5 kg block slides down 15° incline, µk = 0.05. Find acceleration.
  19. A 12 kg crate on 35° incline, µk = 0.15. Compute acceleration.
  20. A 20 kg block slides down 25° incline, friction = 25 N. Find acceleration.
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