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A) Explain how Kepler's $2^{nd}$ law - "The radius vector from the Sun to a planet sweeps out equal areas in equal time intervals" - can be understood in terms of angular momentum conservation.

I know that:

Angular momentum is conserved and therefore $\vec{L}=\vec{r} \times \vec{p}=\vec{r} \times m\vec{v}=constant$ and $L=mrvsin\theta$.

Kepler's $2^{nd}$ law means $\frac{dA}{dt}=constant$

Somehow this comes out to be $dA=(\frac{1}{2})(\frac{L}{m})dt$ but I'm having a hard time getting there.

B) Explain how circular motion can be described as simple harmonic motion.

I know that:

For circular motion $m\vec{a}=\vec{F}_{c}=-m\frac{v^2}{R}\vec{r}=-m\omega^2R\vec{r}$

However, I'm fairly lost on this equation. Where does the negative sign come from, and where does the $\vec{r}$ come from?

A) Explain how Kepler's $2^{nd}$ law - "The radius vector from the Sun to a planet sweeps out equal areas in equal time intervals" - can be understood in terms of angular momentum conservation.

I know that:

Angular momentum is conserved and therefore $\vec{L}=\vec{r} \times \vec{p}=\vec{r} \times m\vec{v}=constant$ and $L=mrv\sin\theta$.

Kepler's $2^{nd}$ law means $\frac{dA}{dt}=constant$

Somehow this comes out to be $dA=(\frac{1}{2})(\frac{L}{m})dt$ but I'm having a hard time getting there.

B) Explain how circular motion can be described as simple harmonic motion.

I know that:

For circular motion $m\vec{a}=\vec{F}_{c}=-m\frac{v^2}{R}\vec{r}=-m\omega^2R\vec{r}$

However, I'm fairly lost on this equation. Where does the negative sign come from, and where does the $\vec{r}$ come from?