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The number of bacteria, \(B\), in a culture
that’s subject to refrigeration can be approximated by the formula:
$$\large B = 300000 \; e^{-0.032 \; t}$$
Where:
\(B\) – number of bacteria
\(t\) – time the culture has been
refrigerated, \(hours\)
Using the given formula, calculates the
number of bacteria in the culture, and display the results after 12, 18, 24, .
. ., 72 hours.
Solution
C++ Code
// Calculating the number of
bacterial Growth in specified Time
// Ref. C++ for Engineers and Scientist, page 170
// Gani, 2025-12-12 #include <iostream> #include <cmath> #include <iomanip> using namespace std;
// function prototype int calcNumBacteria(int
time);
int main()
{ int
const minTime = 12; int
const maxTime = 72; int
const stepTime = 6; cout
<< "\n"; //
Table title cout
<< setw(14) <<
left << "Time (hours)" << setw(18) <<
left << "Number of Bacteria" << endl; //
Calculation for
(int
time =
minTime;
time <=
maxTime;
time +=
stepTime)
{ int
const numBacteria = calcNumBacteria(time); cout
<< setw(8) <<
right << time << setw(18) <<
right << numBacteria << endl;
} return
0;
}
// Given function to calculate the
number of bacteria int calcNumBacteria(int
const time) { const
double A = 300000; // constant const
double n = 0.032; // constant double
const result = A * exp(-n * time); //
Explicitly round the double to the nearest whole number before
// returning the int return
static_cast<int>(round(result));
}
Console Output
Time (hours)Number of Bacteria
12204339
18168643
24139182
30114868
3694801
4278240
4864572
5453292
6043982
6636299
7229958
Feel free to comment for inquiry,
clarification, correction or suggestion for improvement. Drop your email to make
a request to the author.
Disclaimer: The formulas and calculations
presented are for technical reference only. Users must verify accuracy and
ensure compliance with applicable engineering standards, codes, and safety
requirements before practical application.
References
Gary R. Bronson and G.J. Borse. C++ for
Engineers & Scientist. 3rd Ed. Course Technology, Cengage
Learning, Boston, Massachusetts. 2010.
The Mach number is a dimensionless
parameter in fluid dynamics that expresses the ratio between the flow velocity
and the local speed of sound. It is named in honor of Austrian physicist and
philosopher Ernst Mach.
$$M = \frac{u}{c}$$
where: \(M\) – Mach number \(u\) – local flow velocity with respect to
the boundaries \(c\) – speed of sound in the medium
The speed of sound depends on the square
root of the thermodynamic temperature. By definition, \(\text{Mach} \; 1\) corresponds to a
flow velocity \(u\) equal to the local speed of sound. At \(\text{Mach} \; 0.5\), \(u\) is half the
speed of sound (subsonic), while at \(\text{Mach} \; 2.0\), \(u\) is twice the speed of sound
(supersonic). Flow at \(\text{Mach} \; 5.0\) or higher is classified as hypersonic.
At standard atmospheric conditions – dry air
at mean sea level and a temperature of \(15^\circ C\) – the speed of sound is approximately \(340.3 \; m/s\) or \(1,225.1 \; km/h\). The speed of sound is not constant; in gases, it
increases in proportion to the square root of the absolute temperature. Since
atmospheric temperature generally decreases with altitude above sea level, the
speed of sound correspondingly decreases as well.
Flight can be roughly classified in six speed
categories:
When an object moves faster than the speed
of sound (Mach 1), a sudden change in air pressure forms in front of it. This
change, called a shock wave,
spreads backward in a cone shape known as the Mach
cone. The shock wave creates the sonic
boom we hear when a fast aircraft passes by. As the object goes
faster, the cone becomes narrower; just above Mach 1, it looks almost flat.
For further information on the Mach number,
refer to the article “Mach number” in Wikipedia:
The Free Encyclopedia.
Understanding Mach Number Through
Visualization
Figure 1 shows a fighter jet flying at a
supersonic speed of \(Mach \; 2.0\). The cloud that forms around the aircraft at this
speed is called a vapor
cone or shock
collar. It appears as a visible cloud of condensed water vapor
caused by a sudden drop in air pressure and temperature as the jet nears the
speed of sound.
This article provides a Scilab script to
visualize the relative speed of an object and the propagation of sound waves,
as shown in Figure 2, where the object moves at Mach 2.0. The dashed
circle represents the propagation of the shock wave from its center and the
object’s previous position.
For comparison, Figure 3 illustrates an
object moving at the speed of sound along with the resulting shock wave
propagation.
Figure 3. Sonic Speed Simulation (Mach
1.0).
Scilab Code for Figure 2.
// Copyright (C) 2025 - Gani Comia// Date of creation: 19 Sep 2025// Script: Supersonic Speed Simulation and Mach Coneclear;clc;
// primary parameters
M =2; // Mach number (supersonic > 1)
S =1; // speed of sound
mu =asind(S./M); // angle of mach cone
dist =1:10; // object position by a factor of speed of sound
N =length(dist); // number of positions// secondary parameters
a =1:10// object position at x-direction
b =0// object position at y-direction// visualization of object's position and shock wave propagationclf;
g =gcf()
g.figure_size=[700,700]for i =1:N
plot(dist,0,"ko","linewidth",5)
theta =1:360
r(i)=sind(mu).*(N-i)
x(i,:)= r(i)*cosd(theta)+ a(i);
y(i,:)= r(i)*sind(theta)+ b;
plot(x(i,:),y(i,:),"r--")
a1 =legend(["Moving Object","Shock Wave"],with_box=%f)
a1.font_size=2.5endtitle("Supersonic Speed Simulation at Mach 2.0 in 1D", "fontsize",4)xlabel("Object Position / Shock Wave Propagation","fontsize",3.5)ylabel("Shock Wave Propagation","fontsize",3.5)xgrid(color("grey"),1,7)
a2 =xstring(6.5,4,"https://gani-mech-toolbox.blogspot.com")
a2.font_size=2.5// mach cone plot and visualization
x_val =[a(1)+r(1).*cosd(90-mu) a($)]
y_val =[r(1).*sind(90-mu)0]
slope =diff(y_val)./diff(x_val)disp(slope)// straight line using slope-intercept formula, y = mx + b, and// solve for y-intercept, b, at x = 0
y_int = y_val(1)- slope.*x_val(1)disp(y_int)
x_val_new =[0 a($)]
y_val_new_1 =[y_int 0]
y_val_new_2 =[-y_int 0]plot(x_val_new,y_val_new_1,"b-","linewidth",3)plot(x_val_new,y_val_new_2,"b-","linewidth",3)
a3 =xstring(x_val(1),y_val(1),"Mach Cone",mu-6)
a3.font_size=3
a4 =xstring(x_val(1),-y_val(1)-0.7,"Mach Cone",mu-56)
a4.font_size=3
ax =gca()
ax.data_bounds=[0-5 ; 126];
//disp(r)
Feel free to comment for inquiry,
clarification, correction or suggestion for improvement. Drop your email to make
a request to the author.
Disclaimer: The formulas and calculations
presented are for technical reference only. Users must verify the accuracy and
ensure compliance with applicable engineering standards, codes, and safety
requirements before practical application.
Rotating circular machine elements can be modeled as
rotating disks to evaluate stresses. Similar to the theory of thick-walled
cylinders, both tangential and radial stresses are present, but in this case,
they result from inertial forces within the ring or disk. The tangential and
radial stresses are governed by the following conditions.
The outside radius of the disk is significantly larger than
its thickness, \(r_o \geq 10 \, t\).
The ring or disk has a uniform thickness.
The stresses are evenly distributed throughout the
thickness.
In engineering, the von Mises yield
criterion states that yielding begins when the von Mises stress \(\sigma_v\),
derived from the Cauchy stress tensor, equals the yield strength \(\sigma_y\).
The von Mises Equation for a general
state of stress is
For the 2D multi-axial stress, where \(\sigma_3 = 0\) as boundary condition under principal plane stress, the equivalent
von Mises stress, \(\sigma_{\nu}\), becomes
The use of the von Mises criterion as a
yield criterion is only exactly applicable when the material
properties are isotropic, there is no considerable dynamic or unpredictable
loads, and the ratio of the shear yield strength to the tensile yield strength
has the value as in Equation (7).
Critical components suggest a factor of
safety of \(\text{2.5} \leq N_y \leq \text{3.0} \).
Application
A solid disk (Figure 1) is analyzed at
different rotational speeds to determine safe operating limits. The design
factor indicates the maximum speed at which the disk can rotate without
failure. Using the von Mises Yield Criterion, the equivalent stress is
evaluated from the combined tangential and radial stresses. The design factor,
as given by Equation 8, defines the likelihood of structural failure for the
specified parameters.
Assumed Parameters:
Material:Alloy
Steel
Density:7,850 \(kg/m^3\)
Yield Strength:1,250 \(MPa\)
Poisson’s Ratio:0.29
Radius:0.5 \(m\)
Rotational Speed:2500, 5000, 7500, and 10000 \(rpm\)
Figure 1. Rotating Solid Disk (ChatGPT
Image Sep 10, 2025).
Figure 2 presents the equivalent combined
stresses according to the von Mises yield criterion. The stress distribution
varies radially across the disk, reaching its maximum at the center where \(r = 0\).
Figure 2. von Mises Stress from the Point of Analysis along
the Disk Radius.
A contour plot, shown in Figure 3, provides an alternative
visualization of the stress distribution for a given rotational speed.
Figure 3. von Mises Stress Distribution
Figure 4 shows the design factor at various rotational
speeds. The minimum factor of safety is indicated for each case. Rotational
speed of \(7500 \, \text{rpm} \) and above do not satisfy the criteria for critical
components.
Figure 4. Design Factor using von Mises
Stress for Spinning Disk.
Refer to the following Scilab scripts for
figure reproduction.
Scilab Script for Figure 2 and 4
// Copyright (C) 2025 - Gani Comia// Date of creation: 12 Sep 2025// Script: von Mises Yield Criterion in a Spinning Diskclear;clc;
// Primary parameters
rho =7850; // kg/m^3, density of steel
sigmaY =1250; // MPa, yield strength, alloy steel
rpm =[25005000750010000]; // rpm, rotational speed
radps = rpm.*(2*%pi/60); // rad/s, maximum rotational speed
nu =0.29; // Poisson's ration of carbon steel
r_o =0.5; // m, outer radius of cylinder// Secondary parameters
R =linspace(0,r_o,100)
n =length(radps)// von Mises stress distribution for a rotational speedfor i =1:n
// tangential stress, Pa
sigmaT(i,:)=((rho.*radps(i).^2)/8).*((3+nu).*(r_o).^2-(1-3*nu).*(R.^2));
// radial stress, Pa
sigmaR(i,:)=((rho.*radps(i).^2)/8).*((3+nu).*((r_o).^2-(R.^2)));
// von Mises stress, Pa
sigmaV(i,:)=sqrt(sigmaT(i,:).^2+ sigmaR(i,:).^2-(sigmaT(i,:).*sigmaR(i,:)))// von Mises stress, MPa
sigmaV(i,:)= sigmaV(i,:)/1e6end// Plot of von Mises stress distribution for a given rotation speedclf;
scf(0)
f =gcf()
f.figure_size=[800,800]subplot(2,2,1)plot(R,sigmaV(1,:),"g-","linewidth",4.5);
title("von Mises Stress at 2,500 rpm")xlabel("Disk Radius, m","fontsize",3)ylabel("Stress, MPa","fontsize",3)xgrid(color("grey"),1,7)mprintf("Max von Mises #1 : %0.0f\n", max(sigmaV(1,:)))legend("$\LARGE \sigma_{Vmax} = \text{55 MPa}$",with_box=%f)subplot(2,2,2)plot(R,sigmaV(2,:),"m-","linewidth",4.5);
title("von Mises Stress at 5,000 rpm")xlabel("Disk Radius, m","fontsize",3)ylabel("Stress, MPa","fontsize",3)xgrid(color("grey"),1,7)mprintf("Max von Mises #3 : %0.0f\n", max(sigmaV(2,:)))legend("$\LARGE \sigma_{Vmax} = \text{221 MPa}$",with_box=%f)subplot(2,2,3)plot(R,sigmaV(3,:),"b-","linewidth",4.5);
title("von Mises Stress at 7,500 rpm")xlabel("Disk Radius, m","fontsize",3)ylabel("Stress, MPa","fontsize",3)xgrid(color("grey"),1,7)mprintf("Max von Mises #3 : %0.0f\n", max(sigmaV(3,:)))legend("$\LARGE \sigma_{Vmax} = \text{498 MPa}$",with_box=%f)subplot(2,2,4)plot(R,sigmaV(4,:),"r-","linewidth",4.5);
title("von Mises Stress at 10,000 rpm")xlabel("Disk Radius, m","fontsize",3)ylabel("Stress, MPa","fontsize",3)xgrid(color("grey"),1,7)mprintf("Max von Mises #4 : %0.0f\n", max(sigmaV(4,:)))legend("$\LARGE \sigma_{Vmax} = \text{885 MPa}$",with_box=%f)
a1 =xstring(0.25,860,"https://gani-mech-toolbox.blogspot.com")
a1.font_size=2// factor-of-safety calculation and plotfor j =1:n
Ny(j,:)=(sigmaY./sigmaV(j,:))-1endscf(1)
g =gcf()
g.figure_size=[700,700]plot("ln",R,Ny(1,:),"g-","linewidth",4.5)plot("ln",R,Ny(2,:),"m-","linewidth",4.5)plot("ln",R,Ny(3,:),"b-","linewidth",4.5)plot("ln",R,Ny(4,:),"r-","linewidth",4.5)title("Design Factor using von Mises Stress Criterion for Spinning Disk","fontsize",3.5)xlabel("Disk Radius, m","fontsize",3.5)ylabel("Design Factor, Ny","fontsize",3.5)xgrid(color("grey"),1,7)// minimum design factor for a given rotation annotationformat(5)
a2 =xstring(0.005,Ny(1,1),["N (min) = ",string(Ny(1,1))," at 2,500 rpm"])
a2.font_size=3
a3 =xstring(0.005,Ny(2,1),["N (min) = ",string(Ny(2,1))," at 5,000 rpm"])
a3.font_size=3
a4 =xstring(0.005,Ny(3,1),["N (min) = ",string(Ny(3,1))," at 7,500 rpm"])
a4.font_size=3
a5 =xstring(0.005,Ny(4,1)-0.2,["N (min) = ",string(Ny(4,1))," at 10,000 rpm"])
a5.font_size=3// design parameter annotation
a6 ="$N_y = \frac{\sigma_y}{\sigma_v} - 1$"
a7 ="$\sigma_v = \sqrt{\sigma_1^2 + \sigma_2^2 - \sigma_1 \sigma_2}$"
a8 ="$\frac{\sigma_s}{\sigma_y} = \frac{1}{\sqrt{3}} \; , \; \text{assumption}$"
a9 =xstring(0.0625,10,[a6;a7;a8])
a9.font_size=4
a11 =xstring(0.05,8,"https://gani-mech-toolbox.blogspot.com")
a11.font_size=2.5
ax =gca()
ax.data_bounds=[0.0050; 0.525];
Scilab Script for Figure 3
// Copyright (C) 2025 - Gani Comia// Date of creation: 5 Sep 2025// Script: von Mises Stress Distributionclear;clc;
// Primary parameters
rho =7850; // kg/m^3, density of steel
rpm =[25005000750010000]; // rpm, rotation speed
radps =max(rpm.*(2*%pi/60)); // rad/s, maximum rotational speed
nu =0.29; // Poisson's ration of carbon steel
r_o =0.5; // m, outer radius of cylinder// Create meshgrid
n =250;
x =linspace(-r_o, r_o, n);
y =linspace(-r_o, r_o, n);
[X, Y]=ndgrid(x, y);
// Compute radial distance from center
R =sqrt(X.^2+ Y.^2);
// Define stress distribution at maximum rotational speed
sigmaT =zeros(n, n);
sigmaR =zeros(n, n);
sigmaV =zeros(n, n);
for i =1:n
for j =1:n
if R(i,j)<= r_o then// tangential stress
sigmaT(i,j)=((rho.*radps.^2)/8).*((3+nu).*(r_o).^2-(1-3*nu).*(R(i,j)).^2);
// radial stress
sigmaR(i,j)=((rho.*radps.^2)/8).*((3+nu).*((r_o).^2-(R(i,j)).^2));
// von Mises stress
sigmaV(i,j)=sqrt(sigmaT(i,j).^2+ sigmaR(i,j).^2-(sigmaT(i,j).*sigmaR(i,j)));
else
sigmaT(i,j)=%nan; // Outside the circle
sigmaR(i,j)=%nan;
sigmaV(i,j)=%nan;
endendend// Stress distribution using contour plotclf;
f =gcf()
f.figure_size=[700,700]
sigmaV = sigmaV./(1e6)disp(max(sigmaV)); disp(min(sigmaV));
contourf(x, y, sigmaV, 60);
title("von Mises Stress Distribution in Circular Domain at 10K rpm","fontsize",3.5);
xlabel("x-coordinates","fontsize",3)ylabel("y-coordinates","fontsize",3)xgrid(color("grey"),1,7)isoviewon;
f.color_map=jet(64)colorbar(750,900)
ax =gca()
ax.data_bounds=[-0.6-0.6; 0.60.6]
ax.auto_ticks=["on""on"]
Feel free to comment for inquiry,
clarification, correction or suggestion for improvement. Drop your email to make
a request to the author.
Disclaimer: The formulas and calculations
presented are for technical reference only. Users must verify the accuracy and
ensure compliance with applicable engineering standards, codes, and safety
requirements before practical application.
