Physics / Unit 3: Forces and Newton's Laws
3.7 Inclines
Notes
On a ramp at angle $\theta$, split the weight into parts along and perpendicular to the ramp.
- Along the ramp (downhill): $mg\sin\theta$.
- Perpendicular to the ramp: $mg\cos\theta$, balanced by the normal force, so $N = mg\cos\theta$.
- On a frictionless ramp, $a = g\sin\theta$, whatever the mass.
- Example: a 10 kg box on a $30^\circ$ ramp: downhill force $= 98 \times 0.5 = 49 \text{ N}$; $N = 98 \times 0.866 \approx 84.9 \text{ N}$.
Practice
A cart rolls down a frictionless $30^\circ$ ramp. What is its acceleration?
$a = g\sin 30^\circ = 9.8 \times 0.5 = 4.9 \text{ m/s}^2$.
As a ramp is made steeper, what happens to the normal force on a box resting on it?
$N = mg\cos\theta$, and $\cos\theta$ gets smaller as $\theta$ increases.
A 4 kg block sits on a frictionless ramp tilted at $30^\circ$. What force along the ramp pulls it downhill?
$mg\sin\theta = 4 \times 9.8 \times 0.5 = 19.6 \text{ N}$.
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