Problem — Two boys pushing a crate

Problem

Two boys push a crate across a floor with friction: the first one moves it $2\;\mathrm{m}$ at constant speed, the second one moves it $4\;\mathrm{m}$ while accelerating uniformly from rest. Discuss, for each case, what work is done by the pushing force, what work is done by friction, and how the exchanged energy is distributed.

Solution

The energy balance $L_\text{spinta} + L_\text{attr} = \Delta E_\text{cin}$ always holds, with $L_\text{attr} < 0$ (friction opposes the motion) and $|L_\text{attr}| = F_a\cdot d$ the energy that becomes heat.

Case 1 — constant speed ($d = 2\;\mathrm{m}$). If the speed is constant, kinetic energy does not change: $\Delta E_\text{cin} = 0$. Hence $L_\text{spinta} + L_\text{attr} = 0 \quad\Rightarrow\quad L_\text{spinta} = |L_\text{attr}| = F_a\cdot 2\;\mathrm{m}$ The boy must push with a force exactly equal to friction. All the energy he spends is converted into thermal energy (heat from friction between crate and floor): none remains as energy of motion.

Case 2 — accelerated motion from rest ($d = 4\;\mathrm{m}$). Now the crate gains speed, so $\Delta E_\text{cin} > 0$. The push must exceed friction: $L_\text{spinta} = |L_\text{attr}| + \Delta E_\text{cin} \quad\Rightarrow\quad L_\text{spinta} > |L_\text{attr}|$ The energy spent is split into two parts: a share $|L_\text{attr}| = F_a\cdot 4\;\mathrm{m}$ becomes heat, while the surplus $\Delta E_\text{cin}$ becomes kinetic energy of the moving crate.

$\ev{\text{Case 1: all } L_\text{spinta}\to\text{heat};\quad \text{Case 2: } L_\text{spinta}=|L_\text{attr}|+\Delta E_\text{cin}}$

Links

Topics: Work and energy Concepts: Thermal energy from friction · Work done by a force Skills: Bubble-diagram energy balance Objects: Block