Equilibrium of a rigid body

Why a heavy door is easy to open

• Push near the hinge and a door barely moves; push at the handle and it swings easily.
• Same force, different turning effect 转动效应 — the moment 力矩. It's what keeps beams, bridges and see-saws balanced.

Moments

• The moment of a force about a point = force × perpendicular 垂直 distance from the point.
• It measures the turning effect; a bigger distance means a bigger turn.

Equilibrium 平衡 of a beam: take moments about the pivot 支点

Equilibrium of a rigid body 刚体

• A rigid body is in equilibrium when both conditions hold:
  • the resultant force 合力 is zero (it won't accelerate), and
  • the total moment about any point is zero (it won't rotate).
• Weight acts at the centre of mass 质心.

Taking moments about the pivot A removes its (unknown) reaction from the equation.

Worked example — a beam

• A uniform 4 m beam of weight $100$ N is pivoted at A; a force $F$ at B (4 m away) holds it level.
• Weight acts at the centre, 2 m from A. Moments about A: $F \times 4 = 100 \times 2$, so $F = 50$ N.

Toppling 倾倒 and sliding

• On a slope or with a horizontal push, a body may topple (rotate about an edge) or slide (slip).
• It topples when the line of its weight falls outside the base; it slides when the push exceeds limiting friction 最大静摩擦.
• Find the centre of mass of a lamina or a composite body; a rigid body under coplanar forces is in equilibrium when forces and moments both sum to zero.

You've got it

Write from a balancing force of 40 N

• A uniform 6 m beam of weight 80 N is pivoted at one end.
• A force at the other end balances it. Equilibrium needs zero resultant force and zero total moment.