A solid sphere of radius and density floats in water of density

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Fluid statics and the continuum picture

Computational Physics, Ch Where a function vanishes, and what lies between the points

Problem

A solid sphere of radius

R = 0.060\,\mathrm{m}

and density

\rho_s = 700\,\mathrm{kg\,m^{-3}}

floats in water of density

\rho_f = 1000\,\mathrm{kg\,m^{-3}}

. The sphere sinks to a depth

d

(measured from the lowest point of the sphere to the waterline), so the submerged part is a spherical cap of volume

V_{\rm cap}(d) = \dfrac{\pi d^2}{3}(3R - d)

. At equilibrium the weight of the sphere equals the weight of water displaced. (a) Write the equilibrium condition and rearrange it into the form

f(d) = 0

, with

f

given only in terms of

d

R

\rho_s

and

\rho_f

(the factor

g

should cancel). (b) The depth

d

must lie between

0

(sphere just touching the surface) and

2R

(sphere fully submerged). Evaluate

f(0)

and

f(2R)

using the given numbers. (c) State the bracket that these two values establish for the root, and say in one sentence why the physics guarantees a root lies inside it.

Browse the problem library