The orbital angular momentum of a satellite revolving at a distance r from the centre is L . If the distance is increased to 16 r, then the new angular momentum will be

Options

A. `16 L`
B. `64 L`
C. `(L)/(4)`
D. `4 L`

Explanation

To solve the problem of finding the new angular momentum of a satellite when its distance from the center is increased from \( r \) to \( 16r \), we can follow these steps:

### Step 1: Understand the formula for angular momentum
The angular momentum \( L \) of a satellite in orbit is given by the formula:
\[ L = mvr \]
where \( m \) is the mass of the satellite, \( v \) is its orbital velocity, and \( r \) is the distance from the center of the planet.

### Step 2: Express the orbital velocity
The orbital velocity \( v \) of a satellite can be expressed using the gravitational force:
\[ v = \sqrt{\frac{GM}{r}} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the planet.

### Step 3: Substitute the velocity into the angular momentum formula
Substituting the expression for \( v \) into the angular momentum formula gives:
\[ L = m \left(\sqrt{\frac{GM}{r}}\right) r = m \sqrt{GM} \sqrt{r} \]
This shows that angular momentum \( L \) is directly proportional to \( \sqrt{r} \):
\[ L \propto \sqrt{r} \]

### Step 4: Relate the initial and new angular momentum
Let the initial distance be \( r_1 = r \) and the new distance be \( r_2 = 16r \). The initial angular momentum is \( L_1 = L \) and the new angular momentum is \( L_2 \). Since \( L \propto \sqrt{r} \), we can write:
\[ \frac{L_1}{L_2} = \frac{\sqrt{r_1}}{\sqrt{r_2}} \]
Substituting the values gives:
\[ \frac{L}{L_2} = \frac{\sqrt{r}}{\sqrt{16r}} = \frac{\sqrt{r}}{4\sqrt{r}} = \frac{1}{4} \]

### Step 5: Solve for the new angular momentum
From the equation above, we can rearrange it to find \( L_2 \):
\[ L_2 = 4L \]

### Conclusion
Thus, the new angular momentum when the distance is increased to \( 16r \) is:
\[ L_2 = 4L \]