A 1000-kg satellite travels with an orbital speed of 500 m/s around a planet at an orbital raduis of 8000 km. What is the mass of the planet
To find the mass of the planet, we can use the formula for orbital velocity:
v = sqrt(G * M / r),
where:
- v is the orbital velocity,
- G is the gravitational constant (approximately 6.674 × 10^-11 m^3 kg^-1 s^-2),
- M is the mass of the planet, and
- r is the orbital radius.
We know the orbital velocity (v = 500 m/s) and the orbital radius (r = 8000 km = 8000 * 1000 m = 8,000,000 m).
Rearranging the formula, we can solve for the mass of the planet:
M = v^2 * r / G.
Plugging in the values:
M = (500 m/s)^2 * 8,000,000 m / (6.674 × 10^-11 m^3 kg^-1 s^-2).
Now let's calculate it:
M = (500^2 * 8,000,000) / (6.674 × 10^-11) kg.
M = 2,500,000,000,000 / 6.674 × 10^-11 kg.
M ≈ 3.748 * 10^20 kg.
Therefore, the estimated mass of the planet is approximately 3.748 * 10^20 kg.
To find the mass of the planet, we can use the formula for centripetal force. The centripetal force is given by the equation:
Fc = (m * v²) / r
Where Fc is the centripetal force, m is the mass of the satellite, v is the orbital speed, and r is the orbital radius.
In this case, the centripetal force is due to the gravitational force between the satellite and the planet. We can express the gravitational force as:
Fc = (G * m * M) / r²
Where G is the gravitational constant and M is the mass of the planet.
By equating the above two equations, we have:
(G * m * M) / r² = (m * v²) / r
By rearranging the equation, we get:
M = (v² * r) / (G * r²)
Now we can substitute the given values into the equation:
M = ((500 m/s)² * 8,000,000 m) / (6.67430 × 10^-11 m³/kg/s² * (8,000,000 m)²)
By calculating this expression, we can determine the mass of the planet.