A moon of Jupiter has a nearly circular orbit of radius R and an orbit period of T. which of the following expressions gives the mass of Jupiter?

Options

A. `(2piR)/(T)`
B. `(4pi^(2)R)/(T)`
C. `(2piR^(3))/((GT^(2)))`
D. `(4pi^(2)R^(3))/((GT^(2)))`.

Explanation

To find the mass of Jupiter based on the given information about its moon, we can use the principles of gravitational force and centripetal force. Here’s a step-by-step solution:

### Step 1: Understand the Forces Involved
The moon of Jupiter is in a circular orbit, which means that the gravitational force acting on it provides the necessary centripetal force to keep it in that orbit.

### Step 2: Write the Gravitational Force Equation
The gravitational force \( F_g \) acting on the moon can be expressed as:
\[ F_g = \frac{G M m}{R^2} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of Jupiter,
- \( m \) is the mass of the moon,
- \( R \) is the radius of the orbit.

### Step 3: Write the Centripetal Force Equation
The centripetal force \( F_c \) required to keep the moon in a circular orbit is given by:
\[ F_c = \frac{m v^2}{R} \]
where \( v \) is the orbital speed of the moon.

### Step 4: Relate Orbital Speed to the Period
The speed \( v \) of the moon can be expressed in terms of the orbital period \( T \):
\[ v = \frac{2 \pi R}{T} \]
This is because the distance traveled in one complete orbit is the circumference of the circle, \( 2 \pi R \), and \( T \) is the time taken for one complete orbit.

### Step 5: Substitute the Speed into the Centripetal Force Equation
Substituting \( v \) into the centripetal force equation gives:
\[ F_c = \frac{m}{R} \left(\frac{2 \pi R}{T}\right)^2 = \frac{m}{R} \cdot \frac{4 \pi^2 R^2}{T^2} = \frac{4 \pi^2 m R}{T^2} \]

### Step 6: Set Gravitational Force Equal to Centripetal Force
Since the gravitational force provides the centripetal force, we can set the two equations equal to each other:
\[ \frac{G M m}{R^2} = \frac{4 \pi^2 m R}{T^2} \]

### Step 7: Cancel the Mass of the Moon
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{G M}{R^2} = \frac{4 \pi^2 R}{T^2} \]

### Step 8: Solve for the Mass of Jupiter
Rearranging the equation to solve for \( M \):
\[ M = \frac{4 \pi^2 R^3}{G T^2} \]

### Conclusion
The expression that gives the mass of Jupiter is:
\[ M = \frac{4 \pi^2 R^3}{G T^2} \]