Problem — Lifting and lowering a book

Problem

You lift a $1\;\mathrm{kg}$ book from the floor to the table and then bring it back down to the same position. What is the total work you did? And what is the energy spent by your muscles? Explain the difference.

Solution

Total mechanical work. Call $h$ the height of the table. When lifting the book your force is upward and the displacement is upward: positive work $+m\,g\,h$. When lowering it slowly your force is still upward (you support it) but the displacement is downward: negative work $-m\,g\,h$. Adding: $L_\text{tot} = +m\,g\,h - m\,g\,h = 0$ $\ev{L_\text{tot} = 0}$ Consistent with the fact that the book returns to its starting position with zero kinetic energy: the weight force is conservative, and on a closed path the net work is zero.

Muscle energy. The metabolic energy you spend, however, is greater than zero. Muscles consume energy (ATP) even when doing negative work or simply staying under tension: they do not act like a spring that stores and returns energy. The “negative work” of the descent is not recovered as useful energy, but dissipated as heat in the body.

The difference. The mechanical balance on the book is zero (it starts and ends at rest at the same height), but the biological balance is not: the muscles have limited efficiency and in every phase, rising and falling, burn energy that ends up as heat. Zero work on the book, but positive muscular energy.

Collegamenti

Argomenti: Lavoro ed energia Concetti: Lavoro di una forza · Forze conservative Competenze: Impostazione simbolica