Last updated: 2026-06-15

Inclined Plane

A block on a ramp is one of the most useful pictures in mechanics. It shows how a single force — gravity — splits into parts that do different jobs, and it underlies everything from ski slopes and loading ramps to the screw thread and the humble wedge. The key step is to resolve the weight into a component along the slope and one pressing into it. This calculator does that for you, returning the acceleration down the slope along with the normal force, the force along the slope, the friction force and the weight.

Resolving the weight

Gravity always pulls straight down, but on a slope it is convenient to split that pull into two perpendicular directions: along the surface and into it. The part along the slope, $mg\sin\theta$, is what tends to slide the block downhill. The part into the slope, $mg\cos\theta$, is balanced by the surface and equals the normal force $N$. Friction then opposes sliding with up to $\mu N = \mu mg\cos\theta$ of force.

Formula

The block stays at rest as long as the slope component does not exceed the maximum friction, which simplifies to $\tan\theta \le \mu$. Once $\tan\theta > \mu$, it slides and accelerates at $a = g(\sin\theta - \mu\cos\theta)$. With no friction this is just $a = g\sin\theta$.

Worked example

A 5 kg block sits on a 30° ramp with friction coefficient $\mu = 0.2$ (take $g = 9.80665\ \text{m/s}^2$). The components are:

Since $\tan 30^\circ \approx 0.58$ is greater than $\mu = 0.2$, the block slides, accelerating at:

The slip condition

Notice the mass cancels out of the sliding condition $\tan\theta > \mu$. That means whether a block slips depends only on the angle and the friction coefficient, not on how heavy it is — a feather and a brick of the same material let go at the same angle. The threshold angle $\arctan\mu$ is called the angle of repose, the steepest slope on which the object can rest.

Limitations

This model treats the block as a point sliding on a flat ramp, ignoring rotation, rolling and any tendency to tip over. It uses a single friction coefficient; in reality the static value (which sets the slip threshold) is slightly higher than the kinetic value (which sets the acceleration once moving). It also ignores air resistance and assumes the ramp is rigid and straight.