A 2 -kg disk is constrained horizontally but is free to move...

A 2 -kg disk is constrained horizontally but is free to move vertically. The disk is struck from below by a vertical jet of water. The speed and diameter of the water jet are $10 \mathrm{m} / \mathrm{s}$ and $25 \mathrm{mm}$ at the nozzle exit. Obtain a general expression for the speed of the water jet as a function of height, $h .$ Find the height to which the disk will rise and remain stationary.

Key Concepts

Equilibrium Analysis

Equilibrium analysis is used to determine the conditions under which forces balance. In this context, it involves setting the upward force produced by the water jet (derived from momentum transfer) equal to the downward gravitational force (weight) of the disk, thus finding the height at which the disk remains stationary.

Momentum Conservation in Fluid Jets

When a fluid jet changes direction or is deflected by an object, its momentum transfer produces a force on that object. This principle involves calculating the mass flow rate and the change in velocity of the fluid, which is crucial for determining the force exerted by the jet on the disk.

Conservation of Energy (Bernoulli's Principle)

This concept explains that in an ideal, frictionless fluid, the total mechanical energy (kinetic plus potential) remains constant along a streamline. By applying it, one can relate the fluid's speed at different heights with the gravitational potential energy gained, yielding an expression like v(h) = ?(v?² ? 2gh), where v? is the speed at a reference height.

Continuity Equation

The continuity equation states that, for an incompressible fluid in steady flow, the product of the cross?sectional area and the velocity remains constant along the flow path. It is fundamental in relating the flow properties at the nozzle exit to those at any other point in the jet.

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