Vesta, a minor planet in the Asteriod Belt between Mars and Jupiter, has a mean radius of 525 km and mass of 2.59 X 10^20 kg. The Dawn Spacecraft (m = 1240 kg) orbited Vesta in 2011 & 2012.
Find the local acceleration due to gravity at the mean surface of Vesta. What is the fractional change if instead you are at the top of the highest point on the surface, an additional 22 km from the center of gravity. Assume all of Vesta's mass is concentrated at its center of gravity.
g = G M/R^2
here R = 525,000 meters
M = 2.59*10^20
G = 6.67 * 10^-11
for part 2 calculus helps
dg/dr = -2R G M /R^4
so
dg = -2 G M/R^3 dr
dg/g = -2 G M/R^3 (R^2/GM) dr
dg/g = -2 dR/R
here dR/R = 22/525
To find the local acceleration due to gravity at the mean surface of Vesta, we can use the equation for gravitational acceleration:
a = (G * M) / r^2
where:
a is the acceleration due to gravity,
G is the gravitational constant (approximately 6.674 × 10^-11 N m^2/kg^2),
M is the mass of Vesta, and
r is the radius of Vesta.
Plugging in the given values:
M = 2.59 × 10^20 kg
r = 525 km = 525,000 m
a = (6.674 × 10^-11 N m^2/kg^2 * 2.59 × 10^20 kg) / (525,000 m)^2
Calculating this expression will give us the value of the acceleration due to gravity at the mean surface of Vesta.
Now, to find the fractional change in acceleration if you are at the top of the highest point on the surface, an additional 22 km from the center of gravity, we need to calculate the new radius using the mean radius and the additional distance.
New radius = mean radius + additional distance
= 525 km + 22 km
= 547 km = 547,000 m
Then, we can use the same equation for gravitational acceleration:
a_new = (G * M) / r_new^2
where:
a_new is the new acceleration due to gravity,
G is the gravitational constant,
M is the mass of Vesta, and
r_new is the new radius.
Plugging in the known values:
M = 2.59 × 10^20 kg
r_new = 547 km = 547,000 m
a_new = (6.674 × 10^-11 N m^2/kg^2 * 2.59 × 10^20 kg) / (547,000 m)^2
Calculating this expression will give us the value of the new acceleration due to gravity at the top of the highest point on the surface.
To find the fractional change, we can calculate:
Fractional change = (a_new - a) / a
Substitute the respective values of a_new and a in the equation and calculate to find the fractional change.