Properties of Friction — Question 4

Question 4

Two acceleration measurements isolate friction and an unknown pull

Take $g=10$ m/s$^2$. Use ideal dry friction, $|f_s|\le\mu_sN$ and $|f_k|=\mu_kN$. A 4 kg block slides right on a level floor under the same constant horizontal pull P in two trials. With no added load its acceleration is 3 m/s$^2$. With a firmly attached 2 kg payload its acceleration is 1 m/s$^2$. Both trials share one kinetic coefficient.

1. Write the two force equations.

2. Infer P and $\mu_k$.

3. Predict acceleration with a 4 kg payload and state what cannot be inferred about starting from rest.

Question 4 - Solution

Two acceleration measurements isolate friction and an unknown pull

(a) Account for both inertia and normal load.

For each trial, $N=Mg$ and $P-\mu_kMg=Ma$. Thus $$\boxed{P-40\mu_k=12\ \mathrm N},\qquad\boxed{P-60\mu_k=6\ \mathrm N}.$$ The payload changes both the normal force and inertial mass.

(b) Eliminate the common pull.

Subtract the equations to eliminate the common pull: $$(P-40\mu_k)-(P-60\mu_k)=12-6\Longrightarrow20\mu_k=6.$$ $$\boxed{\mu_k=.30},\qquad\boxed{P=12+40(.30)=24\ \mathrm N}.$$

(c) Predict only within the tested regime.

A 4 kg payload makes the total mass 8 kg: $$\boxed{a=\frac{24-(.30)(8)(10)}8=0}.$$ The rightward-sliding block coasts at constant speed. Starting from rest would require $P>\mu_sMg$, and these kinetic measurements do not determine $\mu_s$.