Friction and Circular Motion: Quick Revision

A brief note on Friction and Circular Motion. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Static friction matches the push, up to $\mu_s N$. Kinetic friction is $\mu_k N$, and $\mu_k < \mu_s$.
• Friction depends on $N$ and the two surfaces, not on the area of contact.
• On a slope: $mg\sin\theta$ along it, $N = mg\cos\theta$. A body starts to slide when $\tan\theta > \mu_s$.
• Circular motion needs a net force $\dfrac{mv^2}{r}$ toward the centre, supplied by a real force.
• Flat bend: friction does it, so $v_{\max} = \sqrt{\mu_s rg}$. Banked bend: the tilted normal reaction helps.

The facts at a glance

• Sliding down a rough slope: $a = g(\sin\theta - \mu_k\cos\theta)$
• Pushed up a rough slope: deceleration $g(\sin\theta + \mu_k\cos\theta)$
• Angle of repose: $\tan\theta_r = \mu_s$
• Banked, no friction: $\tan\theta = \dfrac{v^2}{rg}$
• Banked, fastest with friction: $v^2 = rg\,\dfrac{\mu_s + \tan\theta}{1 - \mu_s\tan\theta}$
• Vertical circle, top: slowest speed $\sqrt{rg}$

A worked example

A bend of radius $60\ \text{m}$ is banked at $12^\circ$, and $\mu_s = 0.35$.

1. Design speed. $v_0 = \sqrt{60\ \text{m} \times 9.8\ \text{m/s}^2 \times \tan 12^\circ} = 11.2\ \text{m/s}$, about $40\ \text{km/h}$.
2. Fastest safe speed. $v^2 = 588\ \text{m}^2/\text{s}^2 \times \dfrac{0.35 + 0.213}{1 - 0.35 \times 0.213} = 588\ \text{m}^2/\text{s}^2 \times \dfrac{0.563}{0.926} = 357\ \text{m}^2/\text{s}^2$, so $v = 18.9\ \text{m/s}$, about $68\ \text{km/h}$.
3. Why faster? Above the design speed the car tends to slide up the bank, so friction acts down it and adds to the force toward the centre.

Practice with new numbers

A program to try

The program finds the fastest safe speed on one banked bend for roads of different grip.

Mistakes to avoid

• Static friction always $\mu_s N$. That is only its largest value.
• $\mu_k$ before checking that the body slides at all.
• An outward "centrifugal" force on a free-body diagram. In the ground's frame there is none.
• km/h in $v^2/r$. Change to m/s first.
• Friction in the wrong direction on a slope or a bend. It always opposes the slipping that would happen without it.

Check yourself

1. A $15\ \text{kg}$ crate is pushed across a floor at a steady speed, with $\mu_k = 0.28$. How hard is the push? ($g = 9.8\ \text{m/s}^2$.)

2. A coin sits $12\ \text{cm}$ from the centre of a turntable. It starts to slip when the turntable reaches $0.9$ turns each second. What is $\mu_s$?

3. Why does spreading sand on an icy road make it safer?

4. What is the fastest safe speed on a flat bend of radius $72\ \text{m}$ with $\mu_s = 0.36$? ($g = 9.8\ \text{m/s}^2$.)

5. Why does a cyclist lean into a bend?