Lesson goal: The Inclined Plane & Friction

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The inclined plane is one of the classic simple machines studied since antiquity by Galileo Galilei and Sir Isaac Newton.

When an object of mass $m$ rests on a ramp tilted at an angle $\theta$, gravity pulls straight down with force $F_g = mg$. We can resolve gravity into two perpendicular components:
  • Downhill component: $F_\parallel = mg \sin\theta$ (pulls the object down along the ramp).
  • Normal component: $F_\perp = mg \cos\theta$ (presses the object firmly into the ramp surface).
The surface pushes back with an equal and opposite normal force $N = mg \cos\theta$.

According to Amontons' laws of friction, the maximum static friction force $F_f$ holding the object in place is: $$F_f \le \mu \cdot N = \mu \cdot mg \cos\theta$$ where $\mu$ is the coefficient of friction between the surfaces.

The Critical Slipping Angle

Motion begins the moment the downhill gravitational pull overcomes friction: $$mg \sin\theta > \mu \cdot mg \cos\theta$$ Notice that the mass $m$ and gravity $g$ cancel out on both sides! Dividing by $\cos\theta$, we get: $$\frac{\sin\theta}{\cos\theta} > \mu \implies \tan\theta > \mu$$ The threshold angle where sliding starts is called the critical angle $\theta_c$: $$\theta_c = \arctan(\mu)$$
  • If $\theta < \theta_c$: Friction wins, and the object stays completely stationary!
  • If $\theta > \theta_c$: Downhill gravity wins, and the object accelerates down the ramp!
add_floor()
add_ramp(260, 220, 340, 16, 25, "#64748b", 0.3)
add_box(160, 120, 36, 24, false, 25, "#38bdf8", 0.1, 0.3)
Move the mouse over a dotted box for more information.

  • $\tan\theta = \mu$: For $\mu = 0.3$, the critical angle is $\theta_c = \arctan(0.3) \approx 16.7^\circ$. Since our ramp is tilted at $25^\circ > 16.7^\circ$, the box slides!
  • Material Friction Coefficients:
    • Teflon / Ice: $\mu \approx 0.04$ (slides on gentle slopes $\approx 2^\circ$).
    • Wood on wood: $\mu \approx 0.30 - 0.40$ (slides above $\approx 17^\circ - 22^\circ$).
    • Rubber on concrete: $\mu \approx 0.80 - 1.00$ (holds on steep slopes up to $\approx 39^\circ - 45^\circ$!).
  • Interactive Physics: Click and drag the sliding block with your mouse while the simulation is running!

Now you try. Change angle = 12 and click Run to watch friction hold the block stationary on the ramp.

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