Laws of motion

Block on a rough incline

Live in your browser, computed by the engine.Open in the playground with sliders and graphs ↗

Intuition

Put a block on a gentle slope and friction holds it still. Tilt the slope more and at some angle it lets go. That angle depends only on the roughness, not on the mass: a heavy block and a light one slip at the same tilt. Once it slides, kinetic friction is a little weaker than the static friction that was holding it, so it does not just creep, it accelerates. Drag the angle slider past the critical value and watch the free-body diagram switch.

The maths

Along the slope gravity contributes $m g \sin\theta$; the normal force is $N = m g \cos\theta$. At rest, static friction equals $m g \sin\theta$ and cannot exceed $\mu_s N$, so the block stays until

$$\tan\theta = \mu_s .$$

Sliding, kinetic friction is $\mu_k N$ and the acceleration is $a = g(\sin\theta - \mu_k \cos\theta)$.

How the engine does it

Friction is decided per contact point every step the way a student is taught: compute the force that would hold the block, compare it with $\mu_s N$, and if it is too much, switch to $\mu_k N$ opposing the motion. The engine reports the actual friction force, so at rest the free-body diagram shows $m g \sin\theta$, not $\mu_s N$. Validated to 1e-10 in both regimes.

Pushing a block on a rough floor

Live in your browser, computed by the engine.Open in the playground with sliders and graphs ↗

Intuition

Push gently and nothing happens, yet something is happening: friction is pushing back with exactly your force. Push harder and friction grows to match, until it cannot. Then the block moves, and because kinetic friction is weaker than the maximum static friction, it accelerates the moment it starts.

The maths

The block moves when $F > \mu_s m g$. Once moving,

$$a = \frac{F - \mu_k m g}{m} .$$

How the engine does it

Below the threshold the engine’s friction readout equals the push to 1e-10, which is the free-body diagram textbooks draw. The slip is reported as a friction_slip event.

Atwood machine

Live in your browser, computed by the engine.Open in the playground with sliders and graphs ↗

Intuition

Two masses hang on a string over a pulley. If they are equal, nothing moves. Make one heavier and it descends while the other rises, both with the same acceleration, because the string keeps them moving together. The string’s tension is the same on both sides and lies between the two weights: more than the lighter weight (it is being pulled up) and less than the heavier (it is being let down).

The maths

Newton’s law on each mass with a common acceleration $a$ and tension $T$:

$$a = \frac{m_2 - m_1}{m_1 + m_2}\, g, \qquad T = \frac{2 m_1 m_2}{m_1 + m_2}\, g .$$

How the engine does it

The string is a rope constraint: an inequality on the path length over the pulley’s tangent geometry. The engine solves for the tension that keeps the path length fixed, and reports it. Validated to round-off against both formulas. Set the masses equal and the acceleration readout is exactly zero.

Block on a table and a hanging mass

Live in your browser, computed by the engine.Open in the playground with sliders and graphs ↗

Intuition

A block on a table is tied to a mass hanging over the edge. The hanging weight pulls the whole system, but it has to accelerate the block as well as itself, so it falls more slowly than free fall. Add table friction and there is a point where the hanging weight is not enough to get the block moving at all.

The maths

With table friction $\mu$, the system moves if $m_2 g > \mu m_1 g$, with

$$a = \frac{m_2 - \mu m_1}{m_1 + m_2}\, g, \qquad T = m_1 (a + \mu g) .$$

The normal force on the block is $m_1 g$ regardless.

How the engine does it

The table is a static polygon with friction, the rope bends over a pulley at its edge. After the hanging mass lands on the floor the rope goes slack, which the event log reports, and a frictionless block simply keeps sliding.