Hi, can someone help me do the following question? (I've cut out some details, leaving the results which might be of help)(adsbygoogle = window.adsbygoogle || []).push({});

Let the vector r represent the displacement from the origin to a moving particle of mass m which is subjected to a force F.

Results which I've been able to arrive at:

[tex]

\mathop H\limits^ \to = \mathop r\limits^ \to \times \mathop p\limits^ \to \Rightarrow \frac{{d\mathop H\limits^ \to }}{{dt}} = m\frac{d}{{dt}}\left( {\mathop r\limits^ \to \times \mathop v\limits^ \to } \right) = \mathop r\limits^ \to \times \mathop F\limits^ \to = \mathop M\limits^ \to

[/tex]

v is velocity, F is force, p = mv is moment.

If F acts towards the origin then [itex]\frac{d}{{dt}}\left( {\mathop r\limits^ \to \times \mathop v\limits^ \to } \right) = \mathop 0\limits^ \to [/itex].

Now here is what I am having trouble with.

Show that [itex]\left\| {\mathop r\limits^ \to \times \mathop v\limits^ \to } \right\|[/itex] is twice the rate at which the area A is swept out by the vector r, that is,

[tex]

\left\| {\mathop r\limits^ \to \times \mathop v\limits^ \to } \right\| = 2\frac{{dA}}{{dt}} = r^2 \frac{{d\theta }}{{dt}}

[/tex]

and hence deduce that, when a mass is subject to a central force directed towards the origin, the vector displacement r moves in a plane and sweeps out equal areas in equal times.

I don't know where to start. I can't visualise what is going on in this problem. Norm of r cross v is the area of a parallelogram...ok...but that doesn't tell me much. As for r, it is arbitrary so I can't try to make a picture. I just don't know what to do here. Can someone help me?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Equal areas in equal times

**Physics Forums | Science Articles, Homework Help, Discussion**