Preparing to anchor a buoy at the edge of a swimming area, a...

Preparing to anchor a buoy at the edge of a swimming area, a worker uses a rope to lower a cubical concrete block, 0.250 $\mathrm{m}$ on each edge, into ocean water. The block moves down at a constant speed of 1.90 $\mathrm{m} / \mathrm{s}$ . You can accurately model the concrete and the water as incompressible. (a) At what rate is the force the water exerts on one face of the block increasing? (b) At what rate is the buoyant force on the block increasing?

Key Concepts

Related Rates in Fluid Dynamics

Related rates problems involve determining the rate at which one quantity changes in relation to another. In the context of fluid dynamics and submerged objects, this concept is used to relate the rate of change in the submerged depth (or area exposed to pressure) with the rate of change in the forces experienced from the fluid. It provides a mathematical tool for dealing with dynamic systems where physical quantities vary with time.

Buoyant Force

Buoyant force is the upward force experienced by an object immersed in a fluid and is equal to the weight of the displaced fluid, according to Archimedes' principle. As more of an object gets submerged, the displaced volume increases, and so does the buoyant force. Understanding this principle is essential for predicting the net forces on objects in fluids and for analyzing stability in floating or submerged bodies.

Force on a Submerged Surface

The force exerted by a fluid on a submerged surface can be calculated by integrating the pressure over the area of the surface. Since hydrostatic pressure varies with depth, different parts of the surface experience different pressures. This concept is key when evaluating how the force changes as the immersed part of an object varies, such as when an object is being lowered into a fluid at a constant rate.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the weight of the fluid above. This pressure increases linearly with depth, as it is determined by the product of the fluid’s density, gravitational acceleration, and the depth below the surface. This concept is central to understanding how forces develop on submerged surfaces and is applicable to a variety of problems involving fluids at rest.

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