A planet is revolving around the Sun in an elliptical orbit. Its closest distance from the sun is `r_(min)`. The farthest distance from the sun is `r_(max)` if the orbital angular velocity of the planet when it is nearest to the Sun `omega` then the orbital angular velocity at the point when it is at the farthest distance from the sun is

Options

A. `(sqrt((r_(min))/(r_(max))))omega`
B. `(sqrt((r_(max))/(r_(min))))omega`
C. `((r_(max))/(r_(min)))^(2)omega`
D. `((r_(min))/(r_(max)))^(2)omega`

Answer

To solve the problem, we need to use the principle of conservation of angular momentum. The angular momentum \( L \) of a planet revolving around the Sun can be expressed as:

\[ L = m \cdot v \cdot r \]

where:
- \( m \) is the mass of the planet,
- \( v \) is the orbital velocity,
- \( r \) is the distance from the Sun.

We can also express the orbital velocity \( v \) in terms of the angular velocity \( \omega \):

\[ v = r \cdot \omega \]

Substituting this into the angular momentum equation gives us:

\[ L = m \cdot (r \cdot \omega) \cdot r = m \cdot r^2 \cdot \omega \]

Since there is no external torque acting on the planet, the angular momentum is conserved. Therefore, we can say:

\[ m \cdot r_{\text{min}}^2 \cdot \omega = m \cdot r_{\text{max}}^2 \cdot \omega' \]

where:
- \( \omega \) is the angular velocity at the closest distance \( r_{\text{min}} \),
- \( \omega' \) is the angular velocity at the farthest distance \( r_{\text{max}} \).

Since the mass \( m \) is constant, we can cancel it from both sides:

\[ r_{\text{min}}^2 \cdot \omega = r_{\text{max}}^2 \cdot \omega' \]

Now, we can solve for \( \omega' \):

\[ \omega' = \frac{r_{\text{min}}^2}{r_{\text{max}}^2} \cdot \omega \]

Thus, the orbital angular velocity at the farthest distance from the Sun is:

\[ \omega' = \frac{r_{\text{min}}^2}{r_{\text{max}}^2} \cdot \omega \]

### Summary of the Solution:
1. Write the expression for angular momentum.
2. Substitute the expression for velocity in terms of angular velocity.
3. Set up the conservation of angular momentum equation.
4. Cancel mass and rearrange to solve for the angular velocity at the farthest distance.