# 2D SHM Question with Ellipse

1. Sep 18, 2010

### NeedPhysHelp8

The problem statement, all variables and given/known data

A particle undergoes simple harmonic motion in both the x and y directions simultaneously. Its x and y coordinates are given by: x=asin(wt) y=bcos(wt) . Show that the quantity x(dy/dt) - y(dx/dt) is constant around ellipse, and what is the physical meaning of this quantity?

The attempt at a solution

Ok so I got the first part of the question:
x(dy/dt) - y(dx/dt) = -abw

Now I have no clue what the meaning of this quantity is? the units I see are m^2/s so is this angular area??? someone help please

2. Sep 19, 2010

### ehild

Think of planetary motion, Kepler's second law.

ehild

3. Sep 19, 2010

### NeedPhysHelp8

Ok great than you ehild! So it's just saying that the same area is covered in equal time all around ellipse.

4. Sep 19, 2010

### D H

Staff Emeritus

That's why this is a bad example. This is not true in this case.

5. Sep 19, 2010

### NeedPhysHelp8

So what's the meaning of x(dy/dt) - y(dx/dt) then? i'm really confused now

6. Sep 19, 2010

### D H

Staff Emeritus
What do you think it might mean? The rules of this forum preclude us from telling you directly. You need to show some work.

7. Sep 19, 2010

### NeedPhysHelp8

Well the units of the constant are m^2/s so if that value is conserved around the ellipse this means the area per unit time is constant when traveling around ellipse which makes sense since ellipse is symmetric. I just don't know why you said Kepler's 2nd Law is a bad example it made perfect sense to me. If I'm wrong, can you point me in a better direction about how to think of this problem.

8. Sep 19, 2010

### ehild

That quantity is constant during the motion, so it is conserved. What conservation laws do you know?

ehild

9. Sep 19, 2010

### NeedPhysHelp8

Oh I just read ahead, gotcha it's conservation of angular momentum! Thanks

10. Sep 19, 2010

### ehild

Your solution is almost quite correct. Well, the quantity in question is the magnitude of the angular momentum divided by the mass. Anyway, the areal velocity is

dA/dt=1/2 [rxv]=1/2(yvx-xvy)ez,

half the vector product of the position vector with the velocity. This is constant both here and for the orbits of planets. The angular momentum is L=m [r x v]. It is conserved when a body moves in a central force field. Gravity is a central force. The force in your problem is also central, as the acceleration is anti-parallel with the position vector.

ehild