A crate of mass m = 1200 kg stands on the open platform of a…

A crate of mass $m = 1200\ \mathrm{kg}$ stands on the open platform of a truck. What is the largest acceleration the truck can have without the crate sliding on the platform? The coefficient of friction is $\mu = 0.3$.

Explanation of Method

Largest acceleration with a crate on a truck.

Step 1: Friction accelerates the crate

Static friction from the platform is the only horizontal force on the crate; it must give the crate the truck's acceleration: $ma \le \mu mg \;\Longrightarrow\; a \le \mu g = 0.3 \cdot 9.81$.

\[a_{\max} \approx 2.94\ \mathrm{m/s^2}\]

$1200\ \mathrm{kg}$ $f \le \mu mg$ $a$ $ma \le \mu mg \Rightarrow a \le \mu g = 2.94\ \mathrm{m/s^2}$

The only thing that makes the crate speed up with the truck is the friction of the platform, pointing forward. Its maximum $\mu mg$ limits the acceleration; the crate's mass cancels out.

Result

\[\boxed{\,a_{\max} \approx 2.94\ \mathrm{m/s^2}\,}\]

Remark

Check. The original result is correct. The mass of the crate does not matter; the same limit applies to braking.