On a distant planet, the acceleration due to gravity is 5.30 m/s2, and the radius is 4600 km. Use the law of gravitation to calculate the mass of this planet.
g=GM/r^2
calculate M
To calculate the mass of the planet using the law of gravitation, we can use the following formula:
\(F = \frac{{G \cdot M \cdot m}}{{r^2}}\)
Where:
- F is the gravitational force between two objects
- G is the gravitational constant (\(6.67 \times 10^{-11}\, \text{Nm}^2/\text{kg}^2\))
- M is the mass of one object (the planet in this case)
- m is the mass of the second object (which we can neglect in this calculation)
- r is the distance between the centers of the two objects (in this case, the radius of the planet)
We know that the acceleration due to gravity on this distant planet is 5.30 m/s², which is the same as the gravitational force per unit mass (F/m). So, we can rearrange the formula to solve for the mass M:
\(F/m = \frac{{G \cdot M}}{{r^2}}\)
Plugging in the known values:
\(5.30\, \text{m/s}^2 = \frac{{(6.67 \times 10^{-11}\, \text{Nm}^2/\text{kg}^2) \cdot M}}{{(4,600,000\, \text{m})^2}}\)
Now, we can solve for M:
\(M = \frac{{(5.30\, \text{m/s}^2) \cdot (4,600,000\, \text{m})^2}}{{6.67 \times 10^{-11}\, \text{Nm}^2/\text{kg}^2}}\)