Weight of mass falling through liquid

Question

• If I place a container of fluid on a scale and drop a non-buoyant mass into the fluid, will the scale read less as long as the mass is in motion downward as compared to when the mass is at rest on the bottom of the container?

Part two:

• Would the shape of the container effect the resultant weight while the mass was in motion (falling)?

Answer 1

Very interesting question. The fall of the mass will have different phases. I will only answer the phase where the mass has attained a constant speed and is falling at that speed inside the liquid. During this time, the weight will be same as combined weight of the liquid, the container and the falling mass. It will be same after the mass settles at the bottom. Just after hitting the bottom, the weight will momentarily go higher assuming the mass does not bounce at the bottom. In case of bouncing, the weight will oscillate as many cycles as number of bounces+1 and then settle at combined weight of the liquid, the container and the mass.

The shape of container will not matter as long as it is wide enough for the mass to freely fall without being too close to the sides of the container

Answer 2

The scale's reading will jump up when the mass impacts the liquid, then gradually decrease to a value larger than the original as the mass decelerates. This means that the scale could read the combined weight of both the liquid and the mass before the mass reaches the bottom if the mass is slowed to a constant velocity by then.

Think of the limiting cases:

1. Right before the mass impacts, the mass and the scale aren't interacting, so the scale should be normal.

2. After the mass impacts, it is being deaccelerated, meaning that the liquid (and therefore the scale) must be providing a repulsive force greater than gravity for the net acceleration to be negative. So the reading on the scale is high.

3. After the mass reaches constant velocity, the liquid & scale must be providing a force equal to that of gravity to keep the net acceleration zero. This net force is less than the force in 2 but greater than 1, so the reading on scale is less than in (2) and greater than in (1).

The exact profile of the reading on the scale would probably be highly non-linear (going from the high point to the medium point), but you could derive a good approximation by modeling the resistive force as a simple $F = - bv$ resistive force and solving for v as a function of time based on some initial conditions and for a resistive constant b. I think you'd get an exponential decay in this case.

The shape of the vessel might influence the resistive force if it's odd, but in general it will depend on the liquid more.