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Astro/EPS C12 (2008) – Mike Wong Problem Set 1 Solution Guide
1 Surface gravity
Problem 1A
First find a formula for g:
F = mobject g
F =
G mplanet mobject r^2
mobject g =
G mplanet mobject r^2
g = G mplanet r^2
For the last column of the table, just divide by the Earth’s surface gravitational acceleration g⊕ = 9.8 m sec−^2.
As an example, here is the calculation for the International Space Station. The other parts of the problem are done the same way. Use the distance to the Earth’s center (350 km + the Earth’s radius) and the mass of the Earth, be- cause the space station itself is too tiny to exert much of a gravitational pull on anything.
r = r⊕ + 350 km = 6370 km + 350 km = 6720 km ×
1000 m 1 km
= 6. 72 × 106 m
mplanet = 5. 97 × 1024 kg
g =
G mplanet r^2
- 67 × 10 −^11 m^3 kg−^1 sec−^2 5. 97 × 1024 kg (6. 72 × 106 )^2 m^2
= 8.82 m sec−^2
% of Earth surface gravity = g/g⊕ = 8.82 m sec−^2 / 9 .8 m sec−^2 = 90%
Location
Distance to center of planet/object (m)
Mass of planet/ object (kg)
Gravitational acceleration
(m s–2)
(% of Earth surface gravity) Surface of the Moon 1.74 106 7.35 1022 1.6 17% Surface of Pluto 1.18 106 (p.75) 1.3 1022 0.62 6.4% Surface of Ceres 4.57 105 9.4 1020 0.3 3% Surface of Enchilladus 5.4 105 1.1 1021 0.25 2.6% International Space Station orbiting 350 km above Earth’s surface
6.7 106 5.97 1024 8.8 90%
Problem 1B “Weight is a measure of gravitational force” means we can use the F from part 1A as the weight. Then:
F⊕ = mobject g⊕ FPluto = mobject gPluto
FPluto F⊕
mobject gPluto mobject g⊕
FPluto = F⊕ gPluto g⊕
= 150 lbs × 6 .4% = 9.53 lbs
Problem 1C Solve the equation for g:
g = F/m = 150 lbs/150 lbs = 1
So the answer is just g = 1, which is valid only at the Earth’s surface. So g is actually a convenient unit of acceleration, used for measuring accelerations felt by astronauts during launch, by roller-coaster riders, and by football tacklers.
Problem 1D The astronauts experience a zero-gravity environment because they are in a kind of free-fall. The 0.9 g acceleration at the space station is what keeps it in orbit around the Earth. Astronauts train for zero-g environments by riding in an airplane called the “vomit comet,” which goes up high, then dives down. During the descent, passengers feel weightless because they’re free-falling.
