Question

Choose the correct option(s) for revolution of moon about earth.
Assume the terms to have their usual meanings.

• A. $a\propto \frac{v^2}{R}$
• B. $a\propto \frac{1}{R^2}$
• C. $a\propto \frac{R}{T^2}$
• D. $v\propto \frac{R}{T}$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $a\propto \frac{v^2}{R}$
B $a\propto \frac{1}{R^2}$
C $a\propto \frac{R}{T^2}$
D $v\propto \frac{R}{T}$

When writing proportionalities, it is a good practice to analyze which parameters are constant and then, work on them. For the given question and options, all parameters are variables except the mass of the moon and the mass of the earth.

From the inverse square law,
$a\propto \frac{1}{R^2}$

Centripetal acceleration of the moon is given by:
$a=\frac{v^2}{R}.....\left(i\right)$
$⟹a\propto \frac{v^2}{R}$
Note- The above relations can be used to find relations between $v$ and $R$ for the moon.

Speed of moon in its orbit is given by:
$v=\frac{2\pi R}{T}.....\left(ii\right)$

Substituting $\left(ii\right)$ in $\left(i\right)$, we get:
$a=\frac{(2\pi R/T{)}^2}{R}$
$a=\frac{4{\pi }^{2}R}{T^2}$
$a\propto \frac{R}{T^2}$