When a wooden block is placed on an inclined plane, three main forces act on it: the weight (mg) downward, the normal force (N = mg cosθ) perpendicular to the incline, and the friction force along the incline. The block remains stationary (in equilibrium) if the friction force is sufficient to balance the gravitational component along the incline (mg sinθ). The block begins to slide when the angle θ exceeds the angle of friction.
Normal force on a block on inclined plane: N = mg cosθ.
Gravitational component along incline (pulling block down): mg sinθ.
Block stays in equilibrium when mg sinθ ≤ μmg cosθ, i.e., tanθ ≤ μ.
Angle of repose θ_c: tanθ_c = μₛ (coefficient of static friction).
Block slides when angle θ exceeds the angle of repose.
Acceleration when sliding: a = g(sinθ − μ_k cosθ).
Kinetic friction is less than static friction (μ_k < μₛ).
For a block of mass m on an incline of angle θ:
Weight: W = mg (vertically downward)
Normal Force: N = mg cosθ (perpendicular to incline surface, away from surface)
Friction Force: f ≤ μN = μmg cosθ
Equilibrium condition: mg sinθ = f (friction = gravitational component along incline) mg sinθ ≤ μₛmg cosθ tanθ ≤ μₛ
Critical angle (angle of repose) θ_c: tanθ_c = μₛ (coefficient of static friction)
Decision table:
| Condition | Outcome |
|---|---|
| θ < θ_c (tanθ < μₛ) | Block remains stationary |
| θ = θ_c (tanθ = μₛ) | Block on verge of sliding |
| θ > θ_c (tanθ > μₛ) | Block slides down |
When the block slides, kinetic friction applies: f_k = μ_k mg cosθ
Net force along incline (downward) = mg sinθ − μ_k mg cosθ Acceleration a = g(sinθ − μ_k cosθ)
Note: μ_k < μₛ (kinetic friction < static friction)
When the block slides down a distance d along the incline:
Height descended: h = d sinθ
Energy analysis:
Final velocity after sliding distance d from rest: v² = u² + 2ad v² = 0 + 2 × g(sinθ − μ_k cosθ) × d v = √[2gd(sinθ − μ_k cosθ)]
Three forces act: (1) Weight mg downward (with components mg sinθ along incline and mg cosθ perpendicular to incline), (2) Normal force N = mg cosθ perpendicular to incline surface, and (3) Friction force f ≤ μmg cosθ along the incline (opposing tendency to slide).
A wooden block starts sliding when the angle of inclination θ exceeds the angle of repose θ_c, where tanθ_c = μₛ (coefficient of static friction). At this point, the gravitational component along the incline (mg sinθ) exceeds the maximum static friction force (μₛmg cosθ).
The normal force on a block of mass m on an inclined plane at angle θ is N = mg cosθ. It is perpendicular to the incline surface and equals the component of the block's weight perpendicular to the surface.
On a frictionless incline at angle θ, the only force along the incline is mg sinθ. By Newton's second law: ma = mg sinθ, so a = g sinθ. For example, at θ = 30°, a = 9.8 × sin30° = 9.8 × 0.5 = 4.9 m/s².
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