Archimedes in an Elevator

A block of wood floats in a glass of water. The glass is placed in an elevator. When the elevator accelerates up, what happens to the block in relation to the water?

A) It floats higher.

B) There is no change.

C) It floats lower.

D) It depends or I can't tell.

Solution: B. One way to look at this situation is to say that apparent "g" has increased from the point of view of the occupants of the elevator. Since this variable is in both the formulas for weight (mg) and buoyant force (ρ_fV_dg), there is no change in the volume displaced and therefore no change in the location of the block in the fluid. This is true but not a rigorous proof.

Another view is from the outside where the system is seen as accelerating. It can be shown that the formula for pressure at depth is P = P₀ + ρ(g+a)d and therefore the formula for the buoyant force is F_b = ρ_fV_d,final(g+a). We also know that before the acceleration, W = F_b,initial so mg = ρ_fV_d,initialg and m = ρ_fV_d,initial. Newton's second law for the accelerating system would then be the following:

F_b - W = ma

Substitute the formulas for weight and mass:

ρ_fV_d,final(g+a) - ρ_fV_d,initialg = (ρ_fV_d,initial)a

A little bit of algebra shows that V_d,final = V_d,initial.

Physics is fun!