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A ball of mass m=0.12 kg is a t rest at the origin of the X axis.

At the moment t0=0 a force parallel to the X axis begins to act upon the ball.

The force varies in time by the expression:

F(t)=6*sin(4*pi*t-pi/6)-0.5.

Numerically aproximate the law of motion x(t) of the ball and plot it for the time interval t=[0,10].

m=0.12

initial_time=0;

final_time=10;

N=1001;

t=linspace(initial_time,final_time,N);

dt=t(2)-t(1);

x(1)=0;

x(2)=0;

for i=2:N-1

x(i+1)=2*x(i)-x(i-1)+dt*dt*(6*sin(4*pi*t(i)-pi/6)-0.5);

end;

plot(t,x);

m=0.12 ;

initial_time=0;

final_time=10;

N=1001;

t=linspace(initial_time,final_time,N);

dt=t(2)-t(1);

The ball at t0 is at position 0 on the X axis, so x0=0

there is no initial speed so x1=x0=0

x(1)=0;

x(2)=0;

for i=2:N-1

x(i+1)=2*x(i)-x(i-1)+dt*dt*(6*sin(4*pi*t(i)-pi/6)-0.5)/m;

end

plot(t,x);

I do not think I got the correct solution to this problem, it seems a little bit weird that the ball is only moving in one direction, shouldn't it oscilate araound an center point given that the function is dependant of sin?

Mark Rzewnicki
on 9 Jan 2020

Here's a basic integration of the system, along with plots:

% mass, initial time, final time

m = 0.12;

t0 = 0;

tf = 3;

% number of time intervals and resulting spacing dt

N = 1001;

t = linspace(t0,tf,N);

dt = t(2) - t(1);

% vectors for storage of position, velocity, and acceleration

x = zeros(1,N);

xdot = zeros(1,N);

xdoubledot = zeros(1,N);

%initial conditions for position, velocity, and acceleration

x(1) = 0;

xdot(1) = 0;

xdoubledot(1) = (6*sin(4*pi*t(1)-pi/6)-0.5)/m;

% numerical integration

for i = 2:1:N

x(i) = x(i-1) + xdot(i-1)*dt;

xdot(i) = xdot(i-1) + xdoubledot(i-1)*dt;

xdoubledot(i) = (6*sin(4*pi*t(i)-pi/6)-0.5)/m;

end

figure(1)

plot(t,xdoubledot)

title('acceleration')

figure(2)

plot(t,xdot)

title('velocity')

figure(3)

plot(t,x)

title('position')

I left the final time at 3 seconds so you can get a better picture of what's going on. I encourage you to play around with the values of t0 and tf a little bit.

Notice that you have a constant term (-0.5) added to a sinusoidal term. What do you expect the results of this "offset" to be?

David Hill
on 9 Jan 2020

F=@(t)6*sin(4*pi*t-pi/6)-0.5;

t=0:.01:10;

a=F(t)/.12;

v=cumtrapz(a);

d=cumtrapz(v);

plot(t,d);

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