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

Options

A. `16L`
B. `64L`
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: Determine the velocity of the satellite at distance \( r \)
The orbital velocity \( v \) of a satellite is determined by the gravitational force and is given by:
\[ v = \sqrt{\frac{GM}{r}} \]
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 when the distance is \( r \):
\[ L = m \left(\sqrt{\frac{GM}{r}}\right) r = m \sqrt{GM} \sqrt{r} = m \sqrt{GMr} \]

### Step 4: Calculate the new velocity at distance \( 16r \)
Now, when the distance is increased to \( 16r \), the new velocity \( v' \) will be:
\[ v' = \sqrt{\frac{GM}{16r}} = \frac{1}{4} \sqrt{\frac{GM}{r}} \]

### Step 5: Substitute the new velocity into the angular momentum formula
Now, substituting this new velocity into the angular momentum formula:
\[ L' = m v' (16r) = m \left(\frac{1}{4} \sqrt{\frac{GM}{r}}\right) (16r) \]
\[ L' = m \left(\frac{1}{4} \sqrt{\frac{GM}{r}}\right) (16r) = 4m \sqrt{GM} \sqrt{r} = 4L \]

### Step 6: Conclusion
Thus, the new angular momentum \( L' \) when the distance is increased to \( 16r \) is:
\[ L' = 4L \]

### Final Answer
The new angular momentum will be \( 4L \).

---