A 4 kg block on a 30° slope with friction — find its acceleration

The question

A 4.0 kg block is released on a 30° incline. The coefficient of kinetic friction between the block and the surface is 0.25. Find the block's acceleration down the slope. Take $g=9.8\frac{m}{s^2}$.

What to notice first

Choose axes along and perpendicular to the SLOPE, not horizontal and vertical. With that choice the acceleration has only one component and the normal force sits entirely on the other axis, which is what makes the algebra collapse to a single line.

Working it through

Resolve the weight. The component along the slope drives the motion; the component perpendicular to it is what the surface pushes back against:

Perpendicular to the slope there is no acceleration, so the normal force balances that component exactly:

Kinetic friction opposes the sliding, and its size is set by that normal force:

Newton's second law along the slope. Every term carries a factor of m, so the mass cancels — the 4.0 kg never enters the arithmetic:

Substitute sin 30° $=0.500$ and $\cos 30$° $=0.866$:

The answer

The mass cancelling is a real result, not a coincidence: on a given slope with a given surface, every block accelerates the same way — which is why a heavy crate and a light one slide down together.

The mistake this one catches

Prefer to watch it? The lesson above walks through every line.