Equilibrium — Question 1

Question 1

Rest at one instant is not enough to establish equilibrium

A $2\,\mathrm{kg}$ block lies at rest at $t=0$ on a frictionless horizontal surface. A constant horizontal force $6\,\mathrm N$ begins acting at that instant. Gravity and the normal cancel.

1. Is the block in equilibrium at $t=0$? Find its velocity after $0.20\,\mathrm s$.

2. What additional observation, beyond an instantaneous photograph, would distinguish rest from sustained static equilibrium?

Question 1 - Solution

Rest at one instant is not enough to establish equilibrium

(a) Derivation and calculation

After the force begins, the net horizontal force is nonzero: $$\sum F_x=ma_x=6\ \mathrm N,\qquad \boxed{a_x=6/2=3\ \mathrm{m/s^2}}.$$ With $v_x(0)=0$ and constant acceleration, $$\boxed{v_x(.20)=v_x(0)+a_x(.20)=.60\ \mathrm{m/s}}.$$ The block is instantaneously at rest but is not in equilibrium under this load.

(b) Result and interpretation

Sustained static equilibrium in the chosen inertial frame requires $$\sum\vec F=\vec0,\qquad\sum\vec\tau=\vec0,\qquad \vec v_C(0)=\vec0,\quad\vec\omega(0)=\vec0.$$ A time sequence can test whether position and orientation remain unchanged under the same conditions. A photograph supplies only an instantaneous position, not a force or acceleration measurement.

Physical check

Here the displacement over $.20$ s is $\tfrac12(3)(.20)^2=.060$ m, demonstrating departure from rest.