Gravitation Ques 1
1 If the angular momentum of a planet of mass $m$, moving around sun in a circular orbit is $L$ about the centre of the sun, its areal velocity is
(Main 2019, 9 Jan I)
(a) $\frac{4 L}{m}$
(b) $\frac{2 L}{m}$
(c) $\frac{L}{2 m}$
(d) $\frac{L}{m}$
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Answer:
Correct Answer: 1.( c )
Solution:
The area covered from $P$ to $P^{\prime}$ is $d A$, which is given by
$ \begin{aligned} d A & =\frac{d \theta}{2 \pi} \times \pi r^2 \\ \Rightarrow \quad d A & =\frac{1}{2} r^2 d \theta \text { or } \frac{d A}{d t}=\frac{1}{2} r^2 \frac{d \theta}{d t} \end{aligned} $
where, $\frac{d A}{d t}=$ a real velocity.
$ \frac{d A}{d t}=\frac{1}{2} r^2 \omega \text { or } \frac{d A}{d t}=\frac{1}{2} r^2 \cdot \frac{L}{m r^2} $
(because angular momentum, $L=m r^2 \omega$ )
$ \frac{d A}{d t}=\frac{L}{2 m} $
BETA
