Free-body diagrams, equilibrium & resolving forces

The big idea: A book rests on a table and doesn't move. Why? Gravity pulls it down, the table pushes up just as hard — the two forces cancel.

Equilibrium: That balance is called equilibrium. Here you'll draw the forces on an object and check they add up to zero.

A force needs two bodies: Every force is an interaction between two bodies: the table pushes on the book, the Earth pulls on the book.

Take one of the two bodies away and that force is gone.

Spot it: Draw only the forces on the object — not the forces it pushes back on.

Equilibrium = the forces balance, so the net force is zero. That can mean staying still or moving at steady speed.

On a slope the normal force tilts with the surface. Draw the forces one at a time:

When the forces do not balance: Add the arrows in each direction. A 30 N pull to the right and 12 N of friction to the left give a resultant force of 18 N to the right.

A resultant that is not zero means the object is not in equilibrium: it speeds up, slows down or changes direction.

A slanted force is hard to add up. The trick is to resolve it — split it into a horizontal part and a vertical part that, together, do the same job. Resolve just means 'break into perpendicular pieces'.

Which is cos, which is sin?: Measure the angle θ from the horizontal.

cos goes with the side next to the angle (the horizontal one); sin goes with the side across from it (the vertical one).

If the angle is given from the vertical instead, swap them.

On a slope, tilt your axes: split the weight along the slope and at right angles to it.

The slope pair: Along the slope: mg sin θ. Into the slope: mg cos θ (θ is the angle of the slope).

Divide by m and the part of g along the slope is g sin θ: on an 8.0° ramp, 9.8 × sin 8.0° = 1.4 m s⁻².

How this is tested — forces in equilibrium are everywhere in Theme A. It comes up two ways:

The classic trap: The sag is how far a rope dips in the middle. A rope pulled nearly straight (a tiny sag) still has to balance the weight with only a tiny vertical part — so the tension becomes huge.

The exam's favourite scene: A bird lands on a washing line and the line barely dips. Watch what that does to the tension:

The equilibrium recipe: Equilibrium means the forces balance, so the net force is zero — and that must be true in each direction on its own.

So left pull = right pull and up pull = down pull.

Resolve every slanted force first, then balance each direction.

Back to that bird. In the exam you don't draw the posts and the line — you redraw the scene as a free-body diagram: the bird becomes a simple box, with one arrow for each force acting on it: