AP Physics 1 FRQ: Loading stones into a floating tub

Qualitative/Quantitative Translation, 8 pointsAbout 21 minutes · 8 points

A rectangular plastic tub of mass M has a flat bottom of area A and vertical sides. It floats upright in a pool of water of density ρ. A student places stones of total mass m in the tub, and the tub floats at rest with its bottom a depth d below the water surface, as shown. The student then replaces the stones with stones of total mass 2m. The tub still floats upright, and no water enters it.

(A) Predict whether the new depth of the tub's bottom is greater than 2d, equal to 2d, or less than 2d. In a clear, coherent, paragraph-length response that may also contain equations, justify your prediction by reasoning about the forces on the tub-and-stones system. Do not derive an equation for d in this part. 3 points

(B) Derive an expression for the depth d of the tub's bottom when it holds stones of mass m, in terms of M, m, A, ρ and physical constants, as appropriate. Begin your derivation by writing a fundamental principle or an equation from the reference information. 3 points