The a) question i managed to solve it !An object with mass m, is moving in the xy-plane and is subject to a force \(\displaystyle \vec{F}=-k \vec{r}\) where \(\displaystyle k\) is a positive constant and \(\displaystyle \vec{r}\) the position vector of the object.

a) Solve the differential equations of the motion of the object, with initial values: for \(\displaystyle t=0\) we have \(\displaystyle \vec{r}(t=0)=(x_0,y_0)\) and \(\displaystyle \vec{v}(t=0)=(v_{0x},v_{oy})\)

b) Which is the condition in order the object to make circular motion

c) Which is the condition in order the object to makerectilinear motionwith slope \(\displaystyle \frac{\pi}{6}\) with respect to x axis.

I solved the differential equations and i found that

\(\displaystyle x(t)=x_0 \cdot cos\left ( \sqrt{\frac{k}{m}}\cdot t \right)+\sqrt{\frac{m}{k}} \cdot v_{0x} \cdot sin\left ( \sqrt{\frac{k}{m}}\cdot t \right)\)

\(\displaystyle y(t)=y_0 \cdot cos\left ( \sqrt{\frac{k}{m}}\cdot t \right)+\sqrt{\frac{m}{k}} \cdot v_{0y} \cdot sin\left ( \sqrt{\frac{k}{m}}\cdot t \right)\)

For the

**b) question**teacher told us to write equations...

I thought that if i take \(\displaystyle x(t)^2+y(t)^2\) that must always equal to something constant. And in \(\displaystyle t=0\) the object was in the position\(\displaystyle (x_0,y_0)\). That means that the magnitude of the position vector is equal to \(\displaystyle \sqrt{x_0^2+y_0^2}\)

Thus, for every t, it has to be \(\displaystyle x(t)^2+y(t)^2=x_0^2+y_0^2\)

But adding the two equations i found in the a) question i can't find a condition in order x(t)^2+y(t)^2 to be indepedent from t.

For the c) question i know that the centripetal acceleration vector has to be equal το 0... only this...

Any help for b) and c)???

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