Equilibrium and friction

What you’ll learn

• Resolve forces for particle equilibrium.
• Distinguish static, limiting and sliding friction.

Understand the idea

A particle is in equilibrium when its resultant force is zero. Resolve along two convenient perpendicular directions, often horizontal and vertical or parallel and perpendicular to a plane. A three-force equilibrium can also be represented by a closed force triangle.

Static friction adjusts to whatever value is needed for equilibrium up to its limiting value $\mu R$. Only at impending motion, or in the usual sliding model, should you set $F=\mu R$. Its direction opposes actual or impending relative motion. An angled applied force can change the normal reaction, so do not automatically substitute $R=mg$.

Key formulae

$$\sum F_x=0,\qquad\sum F_y=0$$

$$0\le F\le\mu R$$

$$F_{\mathrm{lim}}=\mu R$$

$$mg\sin\theta\quad\text{(downslope weight component)}$$

Worked example

A $5\,\mathrm{kg}$ block rests on a horizontal rough surface with $\mu=0.4$. Find the largest horizontal pull consistent with equilibrium. Take $g=9.8\,\mathrm{m\,s^{-2}}$.

1. Vertical equilibrium gives $R=5g=49\,\mathrm N$.
2. At impending motion, friction reaches $\mu R=0.4(49)$.
3. Horizontal equilibrium equates the pull to this limiting friction.

Answer: $19.6\,\mathrm N$.