Two satellities `s_(1)` a & `s_(2)` of equal masses revolves in the same sense around a heavy planet in coplaner circular orbit of radii `R` & `4R`

Options

A. the ratio of period of revolution `s_(1)` and `s_(2)` 1:4
B. their velocities are in the 2:1
C. their angular momentum about the planet are in the ration 2:1
D. the ratio of angular velocities of `s_(2)` w.r.t. `s_(1)` when all three are in the same line is 4:1

Explanation

To solve the problem, we need to analyze the motion of two satellites, \( S_1 \) and \( S_2 \), revolving around a heavy planet in circular orbits of radii \( R \) and \( 4R \) respectively. We will derive the necessary relationships step by step.

### Step 1: Determine the Time Period of Each Satellite

The time period \( T \) of a satellite in a circular orbit is given by the formula:
\[ T = 2\pi \sqrt{\frac{R^3}{GM}} \]
where \( R \) is the radius of the orbit, \( G \) is the gravitational constant, and \( M \) is the mass of the planet.

For satellite \( S_1 \) (radius \( R \)):
\[ T_1 = 2\pi \sqrt{\frac{R^3}{GM}} \]

For satellite \( S_2 \) (radius \( 4R \)):
\[ T_2 = 2\pi \sqrt{\frac{(4R)^3}{GM}} = 2\pi \sqrt{\frac{64R^3}{GM}} = 2\pi \sqrt{64} \sqrt{\frac{R^3}{GM}} = 8 \cdot 2\pi \sqrt{\frac{R^3}{GM}} = 8T_1 \]

### Step 2: Calculate the Ratio of Time Periods

Now, we can find the ratio of the time periods:
\[ \frac{T_1}{T_2} = \frac{T_1}{8T_1} = \frac{1}{8} \]

### Step 3: Determine the Velocity of Each Satellite

The velocity \( V \) of a satellite in a circular orbit is given by:
\[ V = \sqrt{\frac{GM}{R}} \]

For satellite \( S_1 \):
\[ V_1 = \sqrt{\frac{GM}{R}} \]

For satellite \( S_2 \):
\[ V_2 = \sqrt{\frac{GM}{4R}} = \frac{1}{2} \sqrt{\frac{GM}{R}} = \frac{1}{2} V_1 \]

### Step 4: Calculate the Ratio of Velocities

Now, we can find the ratio of the velocities:
\[ \frac{V_1}{V_2} = \frac{V_1}{\frac{1}{2} V_1} = 2 \implies \frac{V_2}{V_1} = \frac{1}{2} \]

### Step 5: Determine the Angular Momentum of Each Satellite

The angular momentum \( L \) of a satellite is given by:
\[ L = mVR \]

For satellite \( S_1 \):
\[ L_1 = mV_1R \]

For satellite \( S_2 \):
\[ L_2 = mV_2(4R) = m\left(\frac{1}{2}V_1\right)(4R) = 2mV_1R \]

### Step 6: Calculate the Ratio of Angular Momenta

Now, we can find the ratio of the angular momenta:
\[ \frac{L_1}{L_2} = \frac{mV_1R}{2mV_1R} = \frac{1}{2} \]

### Step 7: Determine the Angular Velocity of Each Satellite

The angular velocity \( \omega \) is related to the linear velocity by:
\[ \omega = \frac{V}{R} \]

For satellite \( S_1 \):
\[ \omega_1 = \frac{V_1}{R} \]

For satellite \( S_2 \):
\[ \omega_2 = \frac{V_2}{4R} = \frac{\frac{1}{2}V_1}{4R} = \frac{1}{8} \frac{V_1}{R} = \frac{1}{8} \omega_1 \]

### Step 8: Calculate the Ratio of Angular Velocities

Now, we can find the ratio of the angular velocities:
\[ \frac{\omega_2}{\omega_1} = \frac{1}{8} \]

### Summary of Results

1. The ratio of the time periods \( T_1 : T_2 = 1 : 8 \) (not \( 1 : 4 \)).
2. The ratio of the velocities \( V_1 : V_2 = 2 : 1 \) (correct).
3. The ratio of angular momenta \( L_1 : L_2 = 1 : 2 \) (not \( 2 : 1 \)).
4. The ratio of angular velocities \( \omega_2 : \omega_1 = 1 : 8 \) (not \( 4 : 1 \)).