The gravitational field intensity at a height of 150 km above the surface of Uranus is 8.71 N/kg. The radius of Uranus is 2.56 x 107 m.
a. Calculate the mass of Uranus.
b. Calculate the gravitational field intensity at the surface of Uranus.
c. How much would a 100 kg person weigh on the surface of Uranus?
g =8.71 N/kg.
h = 150 000 m=1.5•10^5 m
R= 8.71•10^7 m.
(a) g = G•M/(R+h)².
M = gR²/G,
where the gravitational constant G =6.67•10^-11 N•m²/kg²,
(b) gₒ = G•M/R².
(c) m•gₒ.
To answer these questions, we will use the gravitational formula and Newton's law of universal gravitation:
a. The gravitational field intensity (g) is given as 8.71 N/kg, and the radius of Uranus (r) is given as 2.56 x 10^7 m. We can use the formula for gravitational field intensity:
g = G * (M/r^2)
Where G is the gravitational constant (approximately 6.674 × 10^-11 N m^2/kg^2), M is the mass of Uranus, and r is the distance from the center of Uranus.
To find the mass of Uranus (M), we rearrange the formula:
M = g * r^2 / G
Substituting the given values:
M = (8.71 N/kg) * (2.56 x 10^7 m)^2 / (6.674 × 10^-11 N m^2/kg^2)
Calculating this expression will give us the mass of Uranus.
b. To calculate the gravitational field intensity at the surface of Uranus, we can use the same formula and substitute the radius of Uranus (2.56 x 10^7 m) as the distance from the center:
g(surface) = G * (M/r^2)
Substituting the values into the formula will give us the gravitational field intensity at the surface.
c. To calculate how much a 100 kg person would weigh on the surface of Uranus, we multiply the person's mass by the gravitational field intensity at the surface of Uranus:
Weight = mass * g(surface)
Substituting the values will give us the weight on the surface of Uranus.