Question

In a planetary motion, the areal velocity of the position vector of a planet depends on angular velocity ($\omega$) and distance ($r$) of the planet from the sun. If so, the correct relation for areal velocity is?
$A)\,\,\dfrac{{dA}}{{dt}}\,\, \propto \,\,\omega r\\\ B)\,\,\dfrac{{dA}}{{dt}}\,\, \propto \,\,{\omega ^2}r\\\ C)\,\,\dfrac{{dA}}{{dt}}\,\, \propto \,\,\omega {r^2}\\\ D)\,\,\dfrac{{dA}}{{dt}}\,\, \propto \,\,\sqrt {\omega r}$

Answer 1

This question tests our concept about angular momentum, Planetary motion, areal velocity. We have to use the relation between the areal velocity and the angular velocity.

Complete step by step answer:
The areal velocity in a planetary motion $\left( {\dfrac{{dA}}{{dt}}} \right)$ is given by the following equation,
$\left( {\dfrac{{dA}}{{dt}}} \right) = \dfrac{L}{{2m}}$
Where, $L$is the angular momentum and m is the mass.
The formula of the angular momentum $L$ is given by:
$L = m{r^2}\omega$
If we put the above value of angular momentum in the equation of areal velocity we get,
$\left( {\dfrac{{dA}}{{dt}}} \right) = \dfrac{{m{r^2}\omega }}{{2m}}$
Cancelling out m from the numerator and denominator in the right-hand side, we get
$\left( {\dfrac{{dA}}{{dt}}} \right) = \dfrac{{{r^2}\omega }}{2}$
If we check the proportionality between the above relation we get:
$\dfrac{{dA}}{{dt}}\,\, \propto \,\,\omega {r^2}$
Hence the correct option is (C).
Additional Information:
Areal velocity is connected with angular momentum. For central force fields the area swept out in the plane of the orbit by the planet’s radius vector is proportional to time. The areal velocity is proportional to the angular momentum of the planet. In some problems, the magnitude of the areal velocity is constant while only its direction changes. In some other problems only the component of the areal velocity vector is constant. Conservation of areal velocity is a property of central force motion and within the context of classical mechanics, is equivalent to the conservation of angular momentum.

Note: In this question we should not consider the constants while checking the proportionality and only should consider the variables and their dependency to each other. We should apply correct formulas of angular momentum including the angular velocity. The equations of the planetary motions should be remembered appropriately.