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. `sqt(2Emr^(2))`
D. `sqrt(2Emr)`

Explanation

To find the angular momentum of a satellite revolving in a circular orbit around the Earth, we can follow these steps:

### Step 1: Understand the relationship between gravitational force and centripetal force.
The gravitational force acting on the satellite provides the necessary centripetal force for its circular motion. The gravitational force \( F_g \) is given by:

\[ F_g = \frac{G M m}{r^2} \]

where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( r \) is the radius of the orbit.

### Step 2: Set the gravitational force equal to the centripetal force.
The centripetal force \( F_c \) needed to keep the satellite in circular motion is given by:

\[ F_c = \frac{m v^2}{r} \]

Setting these two forces equal gives us:

\[ \frac{G M m}{r^2} = \frac{m v^2}{r} \]

### Step 3: Cancel the mass of the satellite \( m \) and rearrange the equation.
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):

\[ \frac{G M}{r^2} = \frac{v^2}{r} \]

Multiplying both sides by \( r \):

\[ \frac{G M}{r} = v^2 \]

### Step 4: Express the kinetic energy \( E \) in terms of \( v \).
The kinetic energy \( E \) of the satellite is given by:

\[ E = \frac{1}{2} m v^2 \]

Substituting \( v^2 \) from the previous step:

\[ E = \frac{1}{2} m \left(\frac{G M}{r}\right) \]

### Step 5: Solve for \( v^2 \) in terms of \( E \).
Rearranging the equation gives:

\[ v^2 = \frac{2E}{m} \]

### Step 6: Write the expression for angular momentum \( L \).
The angular momentum \( L \) of the satellite is given by:

\[ L = m v r \]

### Step 7: Substitute \( v \) in terms of \( E \) into the angular momentum equation.
Substituting \( v \) from the kinetic energy expression:

\[ L = m \left(\sqrt{\frac{2E}{m}}\right) r \]

### Step 8: Simplify the expression for angular momentum.
This simplifies to:

\[ L = m r \sqrt{\frac{2E}{m}} = r \sqrt{2Em} \]

### Final Expression:
Thus, the angular momentum \( L \) of the satellite can be expressed as:

\[ L = \sqrt{2Em} \cdot r \]