Show that, if a system moves from one state of rest to another over a certain time interval, then the average of the total external force over this time interval must be zero. An hourglass of mass M stands on a fixed platform which also measures the apparent weight of the hourglass. The sand is at rest in the upper chamber when, at time t = 0, a tiny disturbance causes the sand to start running through. The sand comes to rest in the lower chamber after a time t = τ .

Find the time average of the appararent weight of the hourglass over the time interval [0, τ ].

[The apparent weight of the hourglass is however not constant in time. One can advance an argument that, when the sand is steadily running through, the apparent weight of the hourglass exceeds the real weight!]