The satellite of mass m revolving in a circular orbit of radius r around the earth has kinetic energy E. then, its angular momentum will be

Options

A. `sqrt((E)/(mr^2))`
B. `E/(2mr^2)`
C. `sqrt(2Emr^2)`
D. `sqrt(2Emr)`

Answer

To find the angular momentum of a satellite of mass \( m \) revolving in a circular orbit of radius \( r \) around the Earth, given its kinetic energy \( E \), we can follow these steps:

### Step 1: Understand the relationship between kinetic energy and velocity
The kinetic energy \( E \) of the satellite is given by the formula:
\[ E = \frac{1}{2} mv^2 \]
where \( v \) is the orbital velocity of the satellite.

### Step 2: Solve for velocity \( v \)
From the kinetic energy formula, we can express \( v \) in terms of \( E \):
\[ v = \sqrt{\frac{2E}{m}} \]

### Step 3: Write the formula for angular momentum
The angular momentum \( L \) of an object moving in a circular path is given by:
\[ L = mvr \]
where \( r \) is the radius of the circular orbit.

### Step 4: Substitute \( v \) into the angular momentum formula
Now, substituting the expression for \( v \) from Step 2 into the angular momentum formula:
\[ L = m \left(\sqrt{\frac{2E}{m}}\right) r \]

### Step 5: Simplify the expression for angular momentum
Simplifying the expression, we have:
\[ L = m \cdot r \cdot \sqrt{\frac{2E}{m}} = r \sqrt{2Em} \]

### Step 6: Final expression for angular momentum
To express \( L \) in terms of \( E \) and \( r \), we can rearrange:
\[ L = \sqrt{2Em} \cdot r \]

### Step 7: Check for the relationship with kinetic energy
Since we know that \( E = \frac{1}{2} mv^2 \), we can also express \( L \) in terms of \( E \):
\[ L = \sqrt{2E} \cdot m^{1/2} \cdot r \]

### Conclusion
Thus, the angular momentum \( L \) of the satellite is:
\[ L = \frac{2Em}{v} \]

### Final Answer
The angular momentum of the satellite is:
\[ L = 2Emr \]