Re: Circular motion in SR
- From: mL <mL.beyond@xxxxxxxxxxxxx>
- Date: Sat, 15 Mar 2008 00:29:58 GMT
ram.rachum@xxxxxxxxx wrote:
I'm trying to write the equations for circular motion according to SR
laws of motion. I'm doing some kind of mistakes, because I'm not
getting real solutions to the equations.
This is what's going on: You have an body of mass m0 circling around a
stationary center point that is a distance of r from the body. There
is a force F that attracts the body towards the center point, making
it move in a circle around it. What's the velocity v of the object?
OK, you have a central force field, F_ = f(r)n_ , where the unit
vector n_ points towards a fix point.
Use the equation dE/dt = F_.v_, to show that the energy E and the speed
v = |v_| are constants when the particle moves in a circle r = R.
> I've used these equations:
>
> v=sqrt(a*r)
> a=F/(m0*gamma^3)
> gamma=1/sqrt(1-(v/c)^2)
>
> I keep getting only imaginary solutions for v. What am I doing wrong?
The force-equation is wrong. For cases with constant rest mass m, you
have in general,
F_ = (gamma)ma_ + (gamma)^3 m(v_.a_)v_ /c^2 ,
which for circular motion, r = R, reduces to
f(R) = (gamma)m a, where a = v^2/R,
- note that v_.a_ = 0, i.e. a_ // n_, according to comments above.
/mel
.
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