Forces and equilibrium

Why bridges don't fall down

• Every bridge, every building, every shelf stays put because the forces 力 on it balance.
• Engineers call this equilibrium 平衡 — the vector sum of all forces is zero. Master this and you can analyse any static structure.

Forces and components 分量

• A force is a vector (size + direction). Split it into components: horizontal $F\cos\theta$, vertical $F\sin\theta$.
• A particle is in equilibrium when the forces balance — the vector sum is zero (components add to zero in every direction).

A force splits into F cos theta across and F sin theta up

Friction 摩擦力

• The contact force has the normal reaction 法向反作用力 $R$ (perpendicular) and friction $F$ (along the surface, opposing sliding).
• Friction grows only up to a maximum (limiting friction 最大静摩擦力):

• where $\mu$ is the coefficient of friction 摩擦系数.

The contact force: normal reaction R up and friction F along the surface, with F at most mu R

Worked example — equilibrium on a slope

• A 5 kg block rests on a rough slope at $20^{\circ}$ to the horizontal. Find the friction force.
• Resolve parallel to slope: $F = mg\sin 20^{\circ} = 5 \times 10 \times 0.342 = 17.1$ N.
• Resolve perpendicular: $R = mg\cos 20^{\circ} = 5 \times 10 \times 0.940 = 47.0$ N.

On a slope, resolve the weight into components parallel (mg sin θ) and perpendicular (mg cos θ) to the surface.

Connected particles

• When two particles are connected by a light string over a smooth pulley 滑轮, they have the same acceleration 加速度 and the same tension 张力.
• Write $F = ma$ for each particle, then solve the two equations simultaneously.

Forces on a block on a rough slope: weight, normal reaction, tension and friction

• Friction reaches its maximum in limiting equilibrium, when the body is about to slip.

You've got it

Write from a resultant of 50 N

• A 40 N force and a 30 N force act at right angles.
• Find the resultant.