To find the gravitational force acting on the planet, we can use the formula for gravitational force, which is given by Newton's law of gravitation:
\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
Where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant, approximately \( 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \),
- \( m_1 \) is the mass of the planet,
- \( m_2 \) is the mass of the star (which we need to find using centripetal force),
- \( r \) is the radius of the circular path.
### Step 1: Identify the known values
- Mass of the planet, \( m_1 = 3 \times 10^{26} \, \text{kg} \)
- Radius of the circular path, \( r = 1.5 \times 10^{12} \, \text{m} \)
- Speed of the planet, \( v = 2 \times 10^{6} \, \text{m/s} \)
### Step 2: Calculate the centripetal force
The gravitational force also acts as the centripetal force required to keep the planet moving in a circular path. The formula for centripetal force \( F_c \) is:
\[ F_c = \frac{m_1 \cdot v^2}{r} \]
### Step 3: Substitute the known values into the centripetal force equation
Substituting the values we have:
\[ F_c = \frac{(3 \times 10^{26} \, \text{kg}) \cdot (2 \times 10^{6} \, \text{m/s})^2}{1.5 \times 10^{12} \, \text{m}} \]
### Step 4: Calculate \( v^2 \)
First, calculate \( v^2 \):
\[ v^2 = (2 \times 10^{6})^2 = 4 \times 10^{12} \, \text{m}^2/\text{s}^2 \]
### Step 5: Substitute \( v^2 \) back into the centripetal force equation
Now substituting \( v^2 \) into the equation:
\[ F_c = \frac{(3 \times 10^{26}) \cdot (4 \times 10^{12})}{1.5 \times 10^{12}} \]
### Step 6: Simplify the equation
Now, simplify the equation:
\[ F_c = \frac{12 \times 10^{38}}{1.5 \times 10^{12}} \]
### Step 7: Calculate the final force
Calculating the above expression gives:
\[ F_c = 8 \times 10^{26} \, \text{N} \]
Thus, the gravitational force acting on the planet is:
\[ F = 8 \times 10^{26} \, \text{N} \]
