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

The given problem requires understanding the concept of area velocity (areal velocity) in orbital mechanics and how it is affected by changes in the planet's properties. Let's solve this problem step by step.

1. **Understanding Area Velocity:**
   • Area velocity or areal velocity is defined as the rate at which area is swept by a line joining the planet and the object in orbit. It is a constant for an object in orbit due to the conservation of angular momentum.
   • Mathematically, area velocity \( A \) is given by: \(A = \frac{1}{2} r^2 \omega\\), where \( r \) is the orbital radius, and \( \omega\) is the angular velocity.
2. **Initial Conditions:**
   • Given area velocity \( A_1 = 10 \) km/sec.
   • The planet initially has a mass \( M \) and radius \( R_0 \).
3. **Changes in Planet Properties:**
   • The density of the planet increases by 10%.
   • The radius of the planet increases by 10%.
   • Density, \(\rho = \frac{M}{V}\), where \(V\) is the volume. Initially: \(V_0 = \frac{4}{3} \pi R_0^3\).
   • After the increase, the new volume \(V = \frac{4}{3} \pi (1.1 R_0)^3\) and new radius is \(1.1 R_0\).
4. **Calculating New Mass:**
   • The new density \(\rho_{\text{new}} = 1.1 \rho\).
   • Equate new density and volume to find the new mass: \(\rho_{\text{new}} = \frac{M_{\text{new}}}{V}\).
   • Therefore, \(M_{\text{new}} = 1.1 \rho \cdot \frac{4}{3} \pi (1.1 R_0)^3 = 1.1 M \cdot 1.331 = 1.4641 M \\).
5. **Effect on Area Velocity:**
   • The area velocity remains constant for the same orbital radius and proportionally increases with the increase in mass if observed at the same actual distance. This implies: \(A_{\text{new}} = A_1 \cdot \sqrt{1.4641} \\).
   • Calculate: \(A_{\text{new}} = 10 \times 1.21 = 12.1 \\) km/sec.
6. **Conclusion:**
   • The new area velocity is 12.1 km/sec, which matches the given correct option.

Therefore, the correct answer is 12.1 km/sec.