Sinking Depth

Theory: Sinking Depth of a Spherical Object in a Fluid

Introduction

When a homogeneous spherical object of radius $ R $ and density $ \rho_s $ is placed in a fluid of density $ \rho_f $, it sinks until the buoyant force balances its weight. The goal is to determine the depth $ h $ to which the sphere submerges.

Forces Acting on the Sphere

Gravitational Force (Weight of the Sphere):

\[W = m g = \rho_s V_s g\]

where $ V_s = \frac{4}{3} :nbsphinx-math:`pi `R^3 $ is the total volume of the sphere.
Buoyant Force (Archimedes’ Principle):

\[B = \rho_f V_f g\]

where $ V_f(h) $ is the volume of the sphere submerged in the fluid, given by:

\[V_f(h) = \frac{\pi h^2 (3R - h)}{3}\]

Equilibrium Condition

At equilibrium, the forces balance:

\[B = W\]

Expanding:

\[\rho_f \cdot V_f(h) \cdot g = \rho_s \cdot V_s \cdot g\]

Canceling $ g $:

\[\rho_f V_f(h) = \rho_s V_s\]

Substituting the volume expressions:

\[\rho_f \frac{\pi h^2 (3R - h)}{3} = \rho_s \frac{4}{3} \pi R^3\]

Simplifying:

\[\rho_f h^2 (3R - h) = 4 \rho_s R^3\]

Numerical Solution

Since this equation is nonlinear, it is solved using a numerical root-finding technique such as the Newton-Raphson method or bisection method.

Bisection Method

[1]:

import numpy as np

def submerged_volume_eq(h, R, rho_s, rho_f):
    return rho_f * (h**2 * (3*R - h)) - 4 * rho_s * R**3

def bisection_method(f, a, b, tol=1e-6, max_iter=100):
    if f(a) * f(b) > 0:
        raise ValueError("Function must have opposite signs at a and b")

    for _ in range(max_iter):
        c = (a + b) / 2
        if abs(f(c)) < tol or (b - a) / 2 < tol:
            return c
        if f(c) * f(a) < 0:
            b = c
        else:
            a = c
    return (a + b) / 2  # Return the midpoint if max iterations reached

def find_sinking_depth(R, rho_s, rho_f):
    f = lambda h: submerged_volume_eq(h, R, rho_s, rho_f)
    return bisection_method(f, 0, 2*R)

# Example usage
R = 0.5        # Radius of the sphere (meters)
rho_s = 800    # Density of the sphere (kg/m^3)
rho_f = 1000   # Density of the fluid (kg/m^3)

h = find_sinking_depth(R, rho_s, rho_f)
print(f"The sphere sinks to a depth of {h:.4f} meters.")

The sphere sinks to a depth of 0.7129 meters.

Newton Raphson Method

[4]:

import numpy as np

def submerged_volume_eq(h, R, rho_s, rho_f):
    return rho_f * (h**2 * (3*R - h)) - 4 * rho_s * R**3

def submerged_volume_derivative(h, R, rho_f):
    return rho_f * (6*R*h - 3*h**2)  # Derivative of the function

def newton_raphson_method(f, df, x0, tol=1e-6, max_iter=100):
    x = x0
    for _ in range(max_iter):
        fx = f(x)
        dfx = df(x)
        if abs(fx) < tol:
            return x
        if dfx == 0:
            raise ValueError("Derivative is zero. Newton-Raphson method fails.")
        x -= fx / dfx
    return x

def find_sinking_depth(R, rho_s, rho_f):
    f = lambda h: submerged_volume_eq(h, R, rho_s, rho_f)
    df = lambda h: submerged_volume_derivative(h, R, rho_f)
    return newton_raphson_method(f, df, R)

# Example usage
R = 0.5        # Radius of the sphere (meters)
rho_s = 800    # Density of the sphere (kg/m^3)
rho_f = 1000   # Density of the fluid (kg/m^3)

h = find_sinking_depth(R, rho_s, rho_f)
print(f"The sphere sinks to a depth of {h:.4f} meters.")

The sphere sinks to a depth of 0.7129 meters.

Sectant Method

[5]:

import numpy as np

def submerged_volume_eq(h, R, rho_s, rho_f):
    return rho_f * (h**2 * (3*R - h)) - 4 * rho_s * R**3

def secant_method(f, x0, x1, tol=1e-6, max_iter=100):
    for _ in range(max_iter):
        f_x0 = f(x0)
        f_x1 = f(x1)
        if abs(f_x1) < tol:
            return x1
        if f_x1 - f_x0 == 0:
            raise ValueError("Zero denominator in secant method.")
        x2 = x1 - f_x1 * (x1 - x0) / (f_x1 - f_x0)
        x0, x1 = x1, x2
    return x1

def find_sinking_depth(R, rho_s, rho_f):
    f = lambda h: submerged_volume_eq(h, R, rho_s, rho_f)
    return secant_method(f, 0, 2*R)

# Example usage
R = 0.5        # Radius of the sphere (meters)
rho_s = 800    # Density of the sphere (kg/m^3)
rho_f = 1000   # Density of the fluid (kg/m^3)

h = find_sinking_depth(R, rho_s, rho_f)
print(f"The sphere sinks to a depth of {h:.4f} meters.")

The sphere sinks to a depth of 0.7129 meters.