Gota de champu.Ecuaciones diferenciales
Publicado por JuanMix (1 intervención) el 09/03/2010 23:55:59
Buenas noches,
Tengo que realizar un scritp que me calcule la caida de una gota de champu... no se utilizar las ecuaciones diferenciales y hize poco...
Les pongo el enunciado en ingles que es como lo tengo y lo que fui capaz de hacer, espero que puedan ayudarme..
Bead dropping through Shampoo
In a memorable TV commercial that aired several years ago, a small bead was placed inside a transparent bottle of Prell® shampoo, and allowed to drop ever so slowly through the thick green liquid. The commercial was visually striking because the descent of the bead was so smooth, so gradual, almost hypnotic.
The bead dropping through the shampoo is an example of a body moving through a resistive medium. To take another example, we are all aware of air resistance—you can feel it by sticking your hand out the window of a moving car. Let’s think a bit more carefully about the physics of the shampoo problem.
The motion of the bead is governed by Newton’s law F = ma. If v(t) denotes the bead’s velocity (measured positive downward, for convenience), then Newton’s law becomes
where m is the mass of the bead, g is the acceleration due to gravity, and b is a measure of the viscosity, or frictional resistance, provided by the shampoo. Here, the drag force
–bv is taken to be proportional to velocity, as found experimentally for small objects moving slowly through a highly viscous medium.
1.
Using the Matlab plot the graph v(t) for a few different choices of m and b. The bead is assumed to be released at rest: v(0)=0
2.
The velocity appears to approach a limiting value (known as the terminal velocity). Find a formula for the terminal velocity and compare its value to the result of the simulations.
3.
Find the time required for the bead to reach 50% of the terminal velocity. Does this time increase or decrease as you increase the viscosity b?
m*dv/dt= -bv+mg
ECUACION SIR
function dydt= ecuacionSIR (t,V);
%function dydt= ecuacionSIR (t,y);
%teniendo en cuenta que hay dos ecuaciones:
%es un vector fila, y hay que dárselo en vector columna
global b m g
dydt=(-b*V+m*g)/m;
dydt=dydt';
return
Script
clc;
close all;
clear all;
global b m g
b=0.079; %cte de viscosidad
m=1; %masa en kilos
g=9.8; %m/s2
h=0.01; %paso temporal
V(1)=0; %vel inicial
%valor exacto
[t,V_ex]=ode45('ecuacionSIR',[0,200],V(1));
t2=0:h:200;
V=zeros(size(t2));
ndata=length(t2);
for j=1:ndata-1;
V(j+1)=V(j)-((-b*V(j)+m*g)/m)*h;
end
plot(t2,V,'g-')
hold on
plot(t,V_ex,'r')
Muchas gracias por todo
Tengo que realizar un scritp que me calcule la caida de una gota de champu... no se utilizar las ecuaciones diferenciales y hize poco...
Les pongo el enunciado en ingles que es como lo tengo y lo que fui capaz de hacer, espero que puedan ayudarme..
Bead dropping through Shampoo
In a memorable TV commercial that aired several years ago, a small bead was placed inside a transparent bottle of Prell® shampoo, and allowed to drop ever so slowly through the thick green liquid. The commercial was visually striking because the descent of the bead was so smooth, so gradual, almost hypnotic.
The bead dropping through the shampoo is an example of a body moving through a resistive medium. To take another example, we are all aware of air resistance—you can feel it by sticking your hand out the window of a moving car. Let’s think a bit more carefully about the physics of the shampoo problem.
The motion of the bead is governed by Newton’s law F = ma. If v(t) denotes the bead’s velocity (measured positive downward, for convenience), then Newton’s law becomes
where m is the mass of the bead, g is the acceleration due to gravity, and b is a measure of the viscosity, or frictional resistance, provided by the shampoo. Here, the drag force
–bv is taken to be proportional to velocity, as found experimentally for small objects moving slowly through a highly viscous medium.
1.
Using the Matlab plot the graph v(t) for a few different choices of m and b. The bead is assumed to be released at rest: v(0)=0
2.
The velocity appears to approach a limiting value (known as the terminal velocity). Find a formula for the terminal velocity and compare its value to the result of the simulations.
3.
Find the time required for the bead to reach 50% of the terminal velocity. Does this time increase or decrease as you increase the viscosity b?
m*dv/dt= -bv+mg
ECUACION SIR
function dydt= ecuacionSIR (t,V);
%function dydt= ecuacionSIR (t,y);
%teniendo en cuenta que hay dos ecuaciones:
%es un vector fila, y hay que dárselo en vector columna
global b m g
dydt=(-b*V+m*g)/m;
dydt=dydt';
return
Script
clc;
close all;
clear all;
global b m g
b=0.079; %cte de viscosidad
m=1; %masa en kilos
g=9.8; %m/s2
h=0.01; %paso temporal
V(1)=0; %vel inicial
%valor exacto
[t,V_ex]=ode45('ecuacionSIR',[0,200],V(1));
t2=0:h:200;
V=zeros(size(t2));
ndata=length(t2);
for j=1:ndata-1;
V(j+1)=V(j)-((-b*V(j)+m*g)/m)*h;
end
plot(t2,V,'g-')
hold on
plot(t,V_ex,'r')
Muchas gracias por todo
Valora esta pregunta
0
