Calculate the mass of sum if the mean radius of the earth's orbit is `1.5xx10^(8)km` and `G=6.67xx10^(-11)Nxxm^(2)//kg^(2)`

Options

A. `M=2xx10^(30)kg`
B. `M=3xx10^(30)`
C. `M=2xx10^(15)kg`
D. `M=3xx10^(15)kg`

Explanation

To calculate the mass of the Sun using the mean radius of Earth's orbit and the gravitational constant, we can follow these steps:

### Step 1: Understand the relationship
We know that the gravitational force provides the necessary centripetal force for the Earth's orbit around the Sun. The formula for gravitational force is given by:
\[ F = \frac{G \cdot M \cdot m}{r^2} \]
where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Sun,
- \( m \) is the mass of the Earth,
- \( r \) is the distance between the Earth and the Sun (mean radius of Earth's orbit).

### Step 2: Relate gravitational force to centripetal force
The centripetal force required to keep the Earth in orbit is given by:
\[ F = \frac{m \cdot v^2}{r} \]
where \( v \) is the orbital velocity of the Earth.

### Step 3: Equate the two forces
Setting the gravitational force equal to the centripetal force gives us:
\[ \frac{G \cdot M \cdot m}{r^2} = \frac{m \cdot v^2}{r} \]

### Step 4: Cancel out mass of the Earth
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{G \cdot M}{r^2} = \frac{v^2}{r} \]

### Step 5: Rearrange to find the mass of the Sun
Rearranging the equation gives us:
\[ M = \frac{v^2 \cdot r}{G} \]

### Step 6: Find the orbital velocity
The orbital velocity \( v \) can be expressed in terms of the orbital period \( T \):
\[ v = \frac{2 \pi r}{T} \]
Substituting this into the mass equation gives:
\[ M = \frac{(2 \pi r / T)^2 \cdot r}{G} \]
\[ M = \frac{4 \pi^2 r^3}{G T^2} \]

### Step 7: Substitute values
Now we can substitute the values:
- \( r = 1.5 \times 10^8 \text{ km} = 1.5 \times 10^{11} \text{ m} \) (convert km to m),
- \( G = 6.67 \times 10^{-11} \text{ N m}^2/\text{kg}^2 \),
- \( T = 1 \text{ year} = 3.15 \times 10^7 \text{ seconds} \).

Substituting these into the equation:
\[ M = \frac{4 \pi^2 (1.5 \times 10^{11})^3}{(6.67 \times 10^{-11}) (3.15 \times 10^7)^2} \]

### Step 8: Calculate
Calculating the values:
1. Calculate \( (1.5 \times 10^{11})^3 \).
2. Calculate \( (3.15 \times 10^7)^2 \).
3. Substitute these values into the equation to find \( M \).

### Step 9: Final result
After performing the calculations, we find:
\[ M \approx 2 \times 10^{30} \text{ kg} \]

### Conclusion
The mass of the Sun is approximately \( 2 \times 10^{30} \text{ kg} \).

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