Block on a Frictionless Incline

physicsmechanics/dynamicsdifficulty 2/5#newton#incline#free-body

A block of mass $m$ rests on a frictionless incline that makes an angle $\theta$ with the
horizontal. Find the magnitude of its acceleration down the slope.

mm mgmg NN θθ
Free-body diagram: weight $mg$ and normal force $N$.
Answer & solution

Answer

$ a = g\sin\theta $

Solution

Resolve forces along and perpendicular to the surface. Perpendicular to the incline the
block is in equilibrium, so $N = mg\cos\theta$. Along the incline only gravity contributes,
giving
$$mg\sin\theta = ma \implies a = g\sin\theta.$$