What Does the Constant Quantity x(dy/dt) - y(dx/dt) Represent in 2D SHM?

[NeedPhysHelp8]

Homework Statement

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

 

[ehild]

Think of planetary motion, Kepler's second law.

ehild

 

[NeedPhysHelp8]

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

 

[D H]

Think of planetary motion, Kepler's second law.

Bad example.

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

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

 

[NeedPhysHelp8]

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

 

[D H]

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

 

[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.

 

[ehild]

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

ehild

 

[NeedPhysHelp8]

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

 

[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(yv_x-xv_y)e_z,

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