Force Law for 2D Motion with Const a,b,w: F = F(r)

[vle1]

A body of mass m moves under the influence of a force F in two dimensions. It has an trajectory
r(t) = aCos(wt)x^ + bSin(wt)y^

a = alpha
b = beta
w = omega, they are not a, b, and w in alphabet
x^,y^: vector unit

a,b,w are constant. Find the force law F = F(r) which corresponds to this motion (This trajectory is an ellipse, but not Keplerian ellipse. Newton's Law of Gravitaion is not the force law you seek here).

 

[neutrino]

Hint: Newton's second law of motion.

 

[vle1]

Can you explain more in detail, it still confuses me. How do we derive the fomular of Newton's Second Law F=ma to this kind of fomular

 

[Hootenanny]

Well we have;

$$\vec{F} = m\vec{a}$$
$$\vec{F} = m\frac{d\vec{v}}{dt}$$
$$\vec{F} = m\frac{d^2\vec{r}}{dt^2}$$

Can you go from here?

 

[vle1]

I have to admit that I'm stupid, I know the Newton's Second Law. But how to get to r(t) = aCos(wt)x^ + bSin(wt)y^

 

[neutrino]

What does $$m\frac{d^2\vec{r}}{dt^2}$$ mean to you?

 

[Hootenanny]

My god man! I've forgot my d's! Duly corrected ...