Circular motion: acceleration always perpendicular to velocity

[Fu-Kwun Hwang]

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Acceleration is the rate of the velocity change, i.e. .
Because velocity is a vector, there are many ways to change velocity:
1. change the magnitude of the velocity (change speed) without change it's direction.
2. change direction of the velocity without change it's magnitude: this is the circular motion.
3. change both magnitude and directtion of the velocity.

If the acceleration is in the direction of velocity, the acceleration will change the magnitude the of velocity(change speed).
However, if the acceleration is always perpendicular to the velocity, the velocity will change direction without change it's magnitude. This is the circular motion.
So, for circular motion, the force(acceleration) always perpendicular to it's velocity.

You can drag the velocity to change its direction and magnitude, the magnitude of the force is changed by slider bar.

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[lookang]

under evolution
dvx/dt =  ( fx=calf(0,vx,vy)/m)
dvy/dt = ( fy=calf(1,vx,vy)/m)


custom:
public double calf(int xid,double vx,double vy){
 cta=Math.atan2(vy,vx);
 if(xid==0)return f*Math.cos(cta+pi/2);
 else return f*Math.sin(cta+pi/2);
}





Question:
why is it so complicated, need int xid ?

is it correct to just let
under evolution
dvx/dt =  f/m*Math.cos(cta+pi/2)
dvy/dt = f/m*Math.sin(cta+pi/2)

[Fu-Kwun Hwang]

If you use the following at evolution page:
dvx/dt =  f/m*Math.cos(cta+pi/2)
dvy/dt = f/m*Math.sin(cta+pi/2)

But the above equations need "cta" value during the evolution page.
If you use cta at constraint page, the "cta" will be at t-dt;
You will find much larger calculation error.
It has something to do with how the evolution were done numerically.
The program need to calculate several values between t to t+dt to know how to changed variables to next time step with high preission (You can check out Runge-Kutta's 4-th order method if you are interested in how it was done).


If you do not like xid;
You can also define two functions:
public double calfx(double vx,double vy){
 cta=Math.atan2(vy,vx);
 return f*Math.cos(cta+pi/2);
 else return f*Math.sin(cta+pi/2);
}
public double calfy(double vx,double vy){
 cta=Math.atan2(vy,vx);
 return f*Math.sin(cta+pi/2);
}

combined with
dvx/dt =  (fx=calfx(vx,vy)/m);
dvy/dt = ( fy=calfy(vx,vy)/m);

[juliaamit]

Dear sir,

In this simulation, u assumed that the body is rotating at constant angular speed. So there will not be any force along the tangential direction ( along v). The only force is along the radial directed towards centre.
The force direction is independent even if the body is rotating clockwise or anti clockwise.

Now as the rotating body is subjected to a force  directed towards centre, it should move towards centre according to Newton's 2nd law.

Will u please explain to me.

More over where can i get the answer of Halliday Resnik physics book. if i ask some questions from this book, will u kindly answer me?

Regards,
Amit

[Fu-Kwun Hwang]

Now as the rotating body is subjected to a force  directed towards centre, it should move towards centre according to Newton's 2nd law.

Newton's Law only indicated that acceleration is in the same direction of the net force. F=m a
Acceleration is the rate of velocity change. So it is in the direction of difference between two velocity.

By the way, circular motion or means that it is moving in circle.

[sean1016]

This is very interesting.  I got the files for offline use but it doesn't include the *.java files (after extracting the jar file).

How did you find the perpedicular force in your simulation?  Because if you simulate every one second the direction/distance moved is not your true heading.  Rather, because the path you take is curved the heading is tangent to the point you are at.

Would you mind posting some of the code you used to show finding the perpendicular acceleration?

Thanks. 

[Fu-Kwun Hwang]

The above simulation was created with EJS (all the simulations under category: Easy Java Simulations (2001- )  are the same)

The ejs source code is an xml file.
Assume you have install Easy java simulation: You can open the source with EJS by

Right click inside the simulation region, and select open EJS model .

The perpendicular force is the force required for circular motion

[freddygon]

It was the great information to me.

[himanshuswaraj]

If you use the following at evolution page:
dvx/dt =  f/m*Math.cos(cta+pi/2)
dvy/dt = f/m*Math.sin(cta+pi/2)

But the above equations need "cta" value during the evolution page.
If you use cta at constraint page, the "cta" will be at t-dt;
You will find much larger calculation error.
It has something to do with how the evolution were done numerically.
The program need to calculate several values between t to t+dt to know how to changed variables to next time step with high preission (You can check out Runge-Kutta's 4-th order method if you are interested in how it was done).


If you do not like xid;
You can also define two functions:
public double calfx(double vx,double vy){
 cta=Math.atan2(vy,vx);
 return f*Math.cos(cta+pi/2);
 else return f*Math.sin(cta+pi/2);
}
public double calfy(double vx,double vy){
 cta=Math.atan2(vy,vx);
 return f*Math.sin(cta+pi/2);
}

combined with
dvx/dt =  (fx=calfx(vx,vy)/m);
dvy/dt = ( fy=calfy(vx,vy)/m);

Thanks, Hwang.