Equilibrium — Question 8

Question 8

A small force residual can bound drift over an observation window

A $5\,\mathrm{kg}$ object starts at rest. Throughout a $10\,\mathrm s$ observation, its unknown net force has magnitude at most $0.020\,\mathrm N$. Its orientation is independently held fixed.

1. Give guaranteed upper bounds on speed and displacement magnitude.

2. Does staying within those bounds prove exact static equilibrium?

Question 8 - Solution

A small force residual can bound drift over an observation window

(a) Derivation and calculation

The acceleration is bounded by $$|\vec a(t)|\le F_{\max}/m=.020/5=.004\ \mathrm{m/s^2}.$$ Using the triangle inequality and initial rest, $$|\vec v(t)|=\left|\int_0^t\vec a(s)\,ds\right|\le a_{\max}t,$$ $$|\Delta\vec r(t)|\le\int_0^t|\vec v(s)|\,ds\le\tfrac12a_{\max}t^2.$$ At $t=10$ s, $$\boxed{|v|\le.040\ \mathrm{m/s}},\qquad\boxed{|\Delta r|\le.20\ \mathrm m}.$$

(b) Result and interpretation

Exact static equilibrium requires $$\vec F_{\mathrm{net}}=\vec0,\qquad\vec v=\vec0$$ with orientation also fixed. The observed bounds allow nonzero forces and drift, so they establish only consistency with limited force knowledge, not exact equilibrium.

Physical check

A constant force of maximum magnitude in one direction attains both bounds, so they are sharp.