Friction — Question 2

Question 2

A moving support supplies the required static friction

Take $g=10$ m/s$^2$. A 3 kg crate rests on a truck bed. The truck accelerates right at 2 m/s$^2$. The contact can supply up to 9 N of static friction; no other horizontal force acts on the crate.

1. Find the actual friction if the crate remains fixed on the bed.

2. Find the largest truck acceleration magnitude compatible with sticking.

3. If the truck moves right but accelerates left at 1 m/s$^2$, find friction and explain its direction.

Question 2 - Solution

A moving support supplies the required static friction

(a) Demand common acceleration.

Sticking requires the crate to share the truck acceleration: $$f_s=ma=3(2)=6\ \mathrm N<9\ \mathrm N.$$ Thus $\boxed{f_s=6\ \mathrm N\text{ rightward}}$ is feasible; the actual force is below its maximum.

(b) Apply the available-force bound.

The available static-friction interval imposes $$|ma|\leq f_{s,\max}\Longrightarrow\boxed{|a|\leq\frac93=3\ \mathrm{m/s^2}}.$$

(c) Follow acceleration, not velocity.

With right positive, the new acceleration is $a=-1$ m/s$^2$: $$\boxed{f_s=ma=3(-1)=-3\ \mathrm N}.$$ Friction acts left to slow the crate with the bed. Without it, the crate would slide forward relative to the decelerating truck.