A rocket is used to place a satellite in orbit about a planet. The velocity of the satellite is
534 m/s and the radius of orbit is 1.21×108
m. What is the mass of planet?
mv²/R =GmM/R²
v² =GM/R
M= v²R/G
(the gravitational constant
G =6.67•10⁻¹¹ N•m²/kg²)
To find the mass of the planet, we can use the formula for the centripetal force acting on the satellite:
F = (m * v^2) / r
Where:
F is the force of gravity acting on the satellite,
m is the mass of the planet,
v is the velocity of the satellite,
and r is the radius of the orbit.
The force of gravity acting on the satellite is given by:
F = (G * m * m) / r^2
Where:
G is the gravitational constant.
We can equate the two expressions for force and solve for the mass of the planet.
(G * m * m) / r^2 = (m * v^2) / r
(G * m) / r = v^2
Simplifying and substituting the given values, we have:
(G * m) / (1.21 x 10^8) = 534^2
Solving for m, we get:
m = (534^2 * 1.21 x 10^8) / G
To calculate the value of G, which is the gravitational constant:
G ≈ 6.67 x 10^-11 N (m/kg)^2
Substituting the value of G and solving for m:
m = (534^2 * 1.21 x 10^8) / (6.67 x 10^-11)
m ≈ 1.35 x 10^24 kg
Therefore, the mass of the planet is approximately 1.35 x 10^24 kg.
To find the mass of the planet, we can use the formula for the centripetal acceleration experienced by the satellite:
a = v^2 / r
where:
a = acceleration
v = velocity of the satellite
r = radius of orbit
We can rearrange the formula to solve for the mass of the planet:
m = (v^2 * r) / G
where:
m = mass of the planet
G = gravitational constant (approximately 6.67430 x 10^-11 Nm^2/kg^2)
Now, let's plug in the values given in the problem:
v = 534 m/s
r = 1.21 x 10^8 m
G = 6.67430 x 10^-11 Nm^2/kg^2
Substituting these values into the formula, we have:
m = (534^2 * 1.21 x 10^8) / (6.67430 x 10^-11)
Calculating this expression will give us the mass of the planet.