Force to move a desk

Problem

Estimate the horizontal force needed to push a loaded desk of $80\ \text{kg}$ at constant speed, knowing that the coefficient of kinetic friction between the wooden legs and the parquet floor is $\mu_d \approx 0{,}3$.

Solution

This is a Fermi-style estimate: we are looking for a reasonable order of magnitude, not an exact figure.

Set-up. Pushing “at constant speed” means zero acceleration, so the applied force must simply balance the kinetic friction force. There is no net force: $F = F_a$

Normal reaction. The desk rests on a horizontal floor, so the normal reaction equals the weight: $N = mg = 80 \cdot 9{,}81 \approx 785\ \text{N}$

Kinetic friction force. The force to overcome is: $F = \mu_d\, mg = 0{,}3 \cdot 80 \cdot 9{,}81 = 0{,}3 \cdot 785 \approx 235\ \text{N}$

$\ev{F \approx 235\ \text{N}}$

Practical interpretation. To make sense of the number, let’s compare it with a weight: $235\ \text{N}$ corresponds to the weight of a mass $m_{\text{equiv}} = \frac{F}{g} = \frac{235}{9{,}81} \approx 24\ \text{kg}.$ Pushing that desk at constant speed therefore requires a force equal to that needed to lift about $24\ \text{kg}$: a considerable effort but within a person’s reach, consistent with everyday experience of moving a heavy piece of furniture across parquet.

Links

Topics: Friction Concepts: Kinetic friction Skills: Fermi estimation