Download 9-18 Consider the uniform 31 kg beam held in place by the wall and

g1. Calculate the force of Earth's gravity on a spacecraft 3 Earth radii above the surface
of the Earth, if its mass is 1400 kg.
Take the mass and radius of the Earth to be 5.98 E24 kg and 6.37 E6 m.
Hint:Draw vectors to show the position of the spacecraft with respect to the center of the
Earth and the direction of the gravitational force for the position of the spacecraft in the
figure.
(use figure g1.gif)
Apply Universal Law of gravitation.
r = vector from center of Earth to spacecraft (r = 0 deg from figure
me = Earth mass = 5.98 E24 kg
ms = mass of spacecraft = 1500 kg
F = force on spacecraft due to earth.(F = 180 deg for figure shown) ) this will show
attractive nature of force
1. F = G*me*ms/r^2
2. r = 4*6.37 E6 m
g2. Calculate the acceleration due to gravity on the Moon. The Moon's radius is 1.74 E6
m and its mass is 7.35 E22 kg.
Hint: Draw vectors to represent the position of the arbitrary point P indicated on the
figure with respect to the center of the Moon
(use figure g2.gif)
Use definition of "acceleration due to gravity"
r = vector from surface point {r = 180 deg for figure)
gmoon = vector from surface of Moon inward
m = mass of Moon = 7.35 E22 kg
1. gmoon = G*m/r^2
2. r = 1.74 E6 m
g3. A hypothetical planet has a radius 1.8 times that of the Earth and its mass is 1.5 times
the mass of the Earth. What is the acceleration of gravity near its surface?
Hint: Draw vectors to represent the position of the arbitrary point indicated on the figure
with respect to the center of the Planet. Repeat for Earth. Assume g, the acceleration of
gravity on surface of the Earth is known.
(use figure g3.gif)
Use definition of acceleration due to gravity for planet
Use definition for acceleration of gravity due to Earth
Re = vector from center of Earth to point Q
Rp = vector from center of planet to point P
ae = acceleration of gravity DUE TO Earth at point Q
ap = acceleration of gravity DUE TO Planet at point P
Me = mass of Earth
Mp = mass of planet
1.
2.
3.
4.
5.
ge = G*Me/Re^2
ge - g
gp = G*Mp/Rp^2
Rp = 1.8*Re
Mp = 1.5*Me
g4. Given that the acceleration of gravity on the surface of Mars is 0.38 of what it is on
Earth, and that the radius of Mars is 3400 km, determine the mass of Mars.
Hint: Draw vectors to represent the position of the arbitrary point P indicated on the
figure with respect to the center of .MARS
(Use figure g4.gif)
Use definition of acceleration of gravity on surface of a planet
R1 = position vector from center of Mar to point P (direction = 90 deg)
M1 = mass of Mars
gmars = acceleration of gravity at point P
1. gmars = G*M1/R1^2
2. gmars = 0.38*g
3. R1 = 3400 km
g5. Determine the mass of the Sun using the known value for the period of the Earth's
orbit, 3.156 E7 s , and its distance from the Sun, 1.496 E11 m. Assume a circular orbit.
Hint: Define vectors for the instantaneous velocity, acceleration, relative position of the
Earth with respect to the center of the Sun for the position of the
Earth in the figure. Use N2L to relate gravitational force to acceleration.
(use figure g5)
apply N2L
Use universal law of gravitation
Use expression relating circular motion velocity to acceleration
Use relation between orbital speed and period for uniform circular motion.
Ro = position of Earth with respect to center of Sun (direction = 0 deg)
a = instantaneous acceleration of Earth
Fg = force on Earth due to Sun, gravitational
v = instantaneous velocity of Earth
T = period of Earth's orbit
Ms = mass of Sum
Me = mass of Earth
1. Fg = G*Ms*Me/Ro^2
2. Fg = Me*a
3. Ro = 1.496 E11 m
4. T = 3.156 E7 s
5. F = Me*a
6. a = v^2/Ro
7. T = 2**Ro/v
(one can use magnitude equations or equations in x-direction)