Question

A cart is set rolling across a level table, at the same speed on every trial. If it runs into a patch of sand, the cart exerts on the sand an average horizontal force of and travels a distance of through the sand as it comes to a stop. If instead the cart runs into a patch of flour, it rolls an average of before stopping. What is the average magnitude of the horizontal force the cart exerts on the flour? (a) 2N (b) 3N (c) 6N (d) 18N (e) none of those answers.

Short answer

The correct option is (a) 2N.

Step-by-step solution

Given information

The average horizontal force exerted by the cart on the sand, f = 6N.

The distance traveled by the cart through the sand, d = 6 Cm=0.06 m.

The average distance the cart rolls on a patch of flour before stopping x=18Cm=0.18m.

Energy conservation

The energy of a system is always conserved, and by balancing the energy input and outputs of a system, the unknown energy losses can be determined.

If there are no energy losses in a system, the value of work done is equal to the total energy of the system.

The average horizontal force the cart exerts on the flour

Based on the energy conservation principle, the change in the cart’s mechanical energy is equal to the work done against friction. The value of the change in mechanical energy of the cart is given by the following:

$\begin{array}{l}\Delta E_{\mathrm{mech}}=fd\\ \Delta E_{\mathrm{mech}}=6\mathrm{N}\times 0.06\mathrm{m}\times \frac{1\mathrm{J}}{1\mathrm{N}·\mathrm{m}}\\ \Delta E_{\mathrm{mech}}=0.36\mathrm{J}\end{array}$

Balancing the mechanical energy change for the cart rolling througha patch of flour,

${\mathrm{\Delta E}}_{\mathrm{mech}}=\mathrm{Fx}$

Here, is the average horizontal force the cart exerts ona patch of flour before stopping.

Put the values in the equation:

$\begin{array}{c}0.36\mathrm{J}=F\times 0.18\mathrm{m}\times \frac{1\mathrm{J}}{1\mathrm{N}·\mathrm{m}}\\ F=\frac{0.36\mathrm{J}}{0.18\mathrm{J}/\mathrm{N}}\\ F=2\mathrm{N}\end{array}$

Hence, the average horizontal force the cart exerts on the flour is 2N, and the correct option is (a) 2N.