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Chapter 3: Problem 103

A sphere, of radius \(R,\) is partially immersed, to depth \(d,\) in a liquid of specific gravity SG. Obtain an algebraic expression for the buoyancy force acting on the sphere as a function of submersion depth \(d\). Plot the results over the range of water depth \(0 \leq d \leq 2 R\).

Short Answer

Expert verified
The buoyancy force acting on the sphere as a function of submersion depth \(d\) is given by \(B = SG \rho_w g \frac{1}{3}\pi d^2(3R-d)\). The graph plotting this equation shows that as depth increases from 0 to R, buoyancy force increases, and as depth increases from R to 2R, buoyancy force decreases.

Step by step solution

01

Derive the Formula for Buoyancy

Recall Archimedes' principle, which states that the buoyant force on a submerged or floating object is equal to the weight of the fluid displaced by the object. In mathematical terms, this can be written as \(B = \rho_f g V_d \), where \(B\) is the buoyant force, \(\rho_f\) is the density of the fluid, \(g\) is the acceleration due to gravity, and \(V_d\) is the volume of fluid displaced. In this case, we know that the density of the fluid \(\rho_f\) is the specific gravity SG times the density of water \(\rho_w\), and the volume of fluid displaced \(V_d\) will be the volume of the sphere immersed in the liquid, which changes with depth \(d\). So the formula becomes \(B = SG \rho_w g V_d\).
02

Find the Volume of the Immersed Sphere

The volume \(V_d\) of the fluid displaced by the sphere depends on the depth of immersion \(d\). When a sphere is partially immersed, the volume displaced is a 'cap' of the sphere, which can be calculated as \(V_d = \frac{1}{3}\pi d^2(3R-d)\).
03

Substitute the volume into the Buoyancy formula

Substitute the volume \(V_d\) into the buoyancy equation from step 1 \(B = SG \rho_w g \frac{1}{3}\pi d^2(3R-d)\). This is the required equation for the buoyancy force acting on the sphere as a function of submersion depth.
04

Plot the Results

To plot the results over the range of water depth from 0 to 2R, substitute the depth \(d\) into the equation to calculate the corresponding buoyant force for each value. Plotting a graph with the depth on the x-axis and the buoyancy on the y-axis will show how the buoyancy changes with the depth of immersion.

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Most popular questions from this chapter

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