A satellite moves around the earth in a circular orbit with speed `v`. If `m` is the mass of the satellite, its total energy is

Options

A. `(3)/(4)mv^(2)`
B. `(1)/(2)mv^(2)`
C. `mv^(2)`
D. `-((1)/(2))mv^(2)`

Answer

To find the total energy of a satellite moving in a circular orbit around the Earth, we can follow these steps:

### Step 1: Understand the components of total energy
The total energy (E) of the satellite in orbit is the sum of its kinetic energy (KE) and gravitational potential energy (PE).

\[ E = KE + PE \]

### Step 2: Write the expression for kinetic energy
The kinetic energy of the satellite can be expressed as:

\[ KE = \frac{1}{2} mv^2 \]

where \(m\) is the mass of the satellite and \(v\) is its orbital speed.

### Step 3: Write the expression for gravitational potential energy
The gravitational potential energy of the satellite in orbit is given by:

\[ PE = -\frac{G M m}{r} \]

where \(G\) is the gravitational constant, \(M\) is the mass of the Earth, and \(r\) is the distance from the center of the Earth to the satellite.

### Step 4: Relate orbital speed to gravitational force
For a satellite in a stable circular orbit, the gravitational force provides the necessary centripetal force. Therefore, we can use the formula for orbital speed:

\[ v = \sqrt{\frac{G M}{r}} \]

### Step 5: Substitute the expression for \(v\) into the kinetic energy formula
Now, substituting \(v\) into the kinetic energy formula, we get:

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

### Step 6: Substitute kinetic energy and potential energy into total energy
Now we can substitute the expressions for kinetic and potential energy into the total energy equation:

\[ E = \frac{1}{2} m \frac{G M}{r} - \frac{G M m}{r} \]

### Step 7: Simplify the total energy expression
Combining the terms, we have:

\[ E = \frac{1}{2} m \frac{G M}{r} - \frac{2}{2} \frac{G M m}{r} = \frac{1}{2} m \frac{G M}{r} - \frac{2}{2} m \frac{G M}{r} = -\frac{1}{2} m \frac{G M}{r} \]

### Final Expression for Total Energy
Thus, the total energy of the satellite in orbit is:

\[ E = -\frac{1}{2} \frac{G M m}{r} \]