A solid object of total volume V₀ and density ρ₀ floats in a liquid of density ρ_L, where ρ_L > ρ₀. Which of the following expressions gives the volume of the object that remains above the surface of the liquid?

A sphere of volume V and density ρ_S is completely submerged in a liquid of density ρ_L and held stationary by a vertical string. If ρ_S > ρ_L, which of the following expressions gives the magnitude of the tension in the string?

A small object of mass m and volume V is released from rest at the bottom of a tank filled with a fluid of density ρ. The density of the object is less than ρ, and drag forces are negligible. Which expression gives the magnitude of the object's initial upward acceleration?

A uniform cube of side length L floats at the interface between two immiscible fluids. The top fluid has density ρ₁ and the bottom fluid has density ρ₂. If the cube is submerged such that exactly half of its volume is in each fluid, which expression gives the mass of the cube?

An ideal gas is enclosed in a cylinder with a movable piston. The gas initially has volume $V_0$ and density $\rho_0$. If the piston is moved to compress the gas to a new volume of $\frac{1}{3}V_0$ while keeping the mass constant, which expression represents the new density?

Planet A has mass $M$ and radius $R$. Planet B has mass $4M$ and radius $2R$. If both planets are uniform spheres, which expression correctly predicts the density of Planet B ($\rho_B$) in terms of the density of Planet A ($\rho_A$)?

A copper wire has length $L$, cross-sectional area $A$, and density $\rho$. The wire is stretched until its length is $2L$. Assuming the volume of the copper remains constant during stretching, what is the new density of the wire?

An ideal fluid flows through a horizontal pipe that widens from a radius of $r$ to a radius of $2r$. If the density of the fluid in the narrow section is $\rho_1$, which expression represents the fluid density $\rho_2$ in the wide section?

A container is filled with a volume $V$ of liquid A (density $\rho$) and a volume $V$ of liquid B (density $3\rho$). If the liquids mix completely and their volumes are additive, what is the density of the resulting mixture?

A chemist mixes a mass $M$ of fluid X (density $\rho$) with a mass $M$ of fluid Y (density $3\rho$). Assuming the volumes are additive, which expression represents the density of the resulting mixture?