Introduction to Tensor Calculus, Relativity Homework

[putongren]

I already have the solutions emailed to me from a D H Lawden textbook. I have trouble understanding the solution as the solution is not formatted properly, and the answer seems to be a little too advanced for me. I hope that some one can help me understand the problem.

1. Homework Statement
A particle of mass m is moving in the plane Oxy under the action of a force f. Oxy is an inertial frame. Ox'y' is rotating relative to the inertial frame so that angle x'Ox = $\omega$ . (r,$\theta$)are the polar components of f, (a_r,a_$\omega$) are the polar components of the particle's acceleration relative to Ox'y', v is the particle's speed relative to this frame and $\theta$ is the angle its direction of motion makes with the radius vector in this frame, obtain the equations of motion in the form:

ma_r = f_r +2m$\omega$ sin $\theta$ + mr$\omega$²

ma_r = f_r +2m$\omega$ sin $\theta$ + mr$\omega$²

Deduce that the motion relative to the rotating frame is in accordance with the second law if, in addition to f, following forces are also taken to act on the particle: (i) m$\omega$²r radially outwards (the centrifugal force, (iii) 2m$\omega$v at right angles to the direction of motion (the Coriolis force) (iii) tranversely (The latter force vanishes if the rotation is uniform.)

Homework Equations

Ok. So I'm looking at the solution and I don't understand how they progress from mathematical argument to the next mathematical argument. Maybe I'm weak on polar calculus.

The Attempt at a Solution

I'm going to attach the solution later, since I'll have to scan that specific solution from the big solution set that comes from the book. As I mentioned before, the answer is not formatted properly, but it might still be clear enough for someone with better expertise to examine.

 

[BvU]

Could the angle x'Ox be $\omega t$ ?

Homework Equations

[/B]Ok. So I'm looking at the solution and I don't understand how they progress from mathematical argument to the next mathematical argument. Maybe I'm weak on polar calculus.

That are not relevant equations !

The Attempt at a Solution

I'm going to attach the solution later, since I'll have to scan that specific solution from the big solution set that comes from the book. As I mentioned before, the answer is not formatted properly, but it might still be clear enough for someone with better expertise to examine.

And that isn't an attempt at a solution. You want to show your own work instead of dumping a picture on the folks who try to help you !

 

[vanhees71]

Yeah, start to work out the problem yourself. That helps more than staring at ready solutions!

 

[putongren]

OK.. it's been 3 years... I'm just picked up the book recently and I'm trying again. I made some typos, so will correct them now:
let (r', Θ) = the final position vector in respect to the inertial frame of reference in polar coordinates, t = time arbitrarily elapsed.

I will attempt to convert the problem from using polar coordinates to cartesian coordinates.

since everyone knows x = x_o + v t + .5 a t²,
y coordinate: r' sinΘ = r sin (θ + ω) + v sin (Φ + ω) t + .5 a_rsin a_θt²

x coordinate: r cos (θ + ω) + v cos (Φ + ω) t + .5 a_rcos a_θt²

I realize that I added a lot more variables such as t and (r', Θ), which made the solution more complex. Is it possible to simplify the problem using this method. The solution makes no sense to me.

 

[putongren]

Here is the solution.