Question

For an object revolving around a planet of mass \( M \) and radius \( R_0 \) at a distance \( r \) from the center of the planet. If area velocity of the object is 10 km/sec. Now if the density of the planet increases by +10% and radius of the planet increases by +10%, then find the new area velocity at the same orbital radius.

• A. 12.1 km/sec
• B. 10 km/sec
• C. 15.5 km/sec
• D. 8.5 km/sec

Hint:

The area velocity depends on the radius and mass (or density) of the planet. If the radius and density increase, the area velocity will also increase proportionally.

Answer 1

The Correct Option is A

Step 1: Use the relationship for area velocity.
The area velocity \( v_a \) for an object revolving around a planet is given by: \[ v_a = \frac{dA}{dt} = \text{constant} \times \frac{r^2}{T} \] where \( r \) is the orbital radius and \( T \) is the orbital period. Step 2: Apply the changes in density and radius.
If the density of the planet increases by 10%, then the mass \( M \) increases by 10%, and if the radius of the planet increases by 10%, the area velocity will change. Using the proportional relationship, the new area velocity is: \[ v_a' = 1.1 \times 1.1 \times v_a \] Step 3: Conclusion.
Substituting the given values and calculating, the new area velocity is \( 12.1 \, \text{km/sec} \). Final Answer: \[ \boxed{12.1 \, \text{km/sec}} \]