A beaker of mass mb containing oil of mass mo and density ρo rests on a scale. A block of iron o

step 1 So here in this problem we have to find the equilibrium readings of this scale and this scale an

A beaker of mass mb containing oil of mass mo and density ρo...

A beaker of mass $m_{b}$ containing oil of mass $m_{o}$ and density $\rho_{o}$ rests on a scale. A block of iron of mass $m_{\mathrm{Fe}}$ suspended from a spring scale is completely submerged in the oil as shown in Figure $\mathrm{P} 14.61$ . Determine the equilibrium readings of both scales.

Key Concepts

Force Equilibrium in Fluid Systems

In a state of equilibrium, the sum of all forces on each component of the system is zero. For the submerged block suspended by a spring, the gravitational force acting downward is partially offset by the buoyant force acting upward. This balance explains the reading on the spring scale. Meanwhile, the container with oil experiences additional forces due to the fluid pressure changes caused by the submerged block, and the overall equilibrium of these forces determines the reading on the beaker’s scale.

Scale Readings and Force Transmission

Scales measure the net force applied to them, whether that be the direct weight of an object or the sum of forces transmitted through a fluid medium. In this context, the spring scale measures the actual tension in the string, which amounts to the weight of the block minus the buoyant force from the displaced oil. The beaker’s scale, on the other hand, supports not only the weight of the beaker and oil but also an extra force arising from the interaction between the oil and the submerged block, demonstrating how forces can be redistributed in connected systems.

Archimedes' Principle

This principle states that any object completely or partially immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. In problems involving submerged objects, such as the iron block in oil, Archimedes’ principle is crucial for determining the magnitude of the buoyant force, which in turn influences other forces in the system.

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