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`

Answer

To solve the problem of finding the new orbital 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 Earth.

### Step 2: Determine the relationship between velocity and distance
The orbital velocity \( v \) of a satellite is given by:
\[ v = \sqrt{\frac{GM}{r}} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth. As the distance \( r \) increases, the orbital velocity \( v \) will change.

### Step 3: Calculate the new velocity when the distance is increased to \( 16r \)
When the distance is increased to \( 16r \), the new orbital velocity \( v' \) can be calculated as:
\[ v' = \sqrt{\frac{GM}{16r}} = \frac{1}{4} \sqrt{\frac{GM}{r}} = \frac{1}{4} v \]
This shows that the new velocity is one-fourth of the original velocity.

### Step 4: Substitute the new values into the angular momentum formula
Now, substituting the new values into the angular momentum formula:
\[ L' = mv'r' = m\left(\frac{1}{4}v\right)(16r) = m \cdot 16r \cdot \frac{1}{4}v \]
Simplifying this gives:
\[ L' = 4(mvr) = 4L \]
where \( L \) is the original angular momentum.

### Step 5: 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 \( 4L \).

---