Gravitation Key: :Points per selected answer checked marked in red:
1. On Earth the acceleration due to gravity at the surface is 9.8 m/s2. Enter the mass
and radius of the Earth into the spreadsheet and verify that g at the surface is 9.8
m/s2. What is the acceleration due to gravity 1 complete radius away from the
surface of Earth?
The spreadsheet predicts that the acceleration due to gravity is 2.45 m/s2. +1
2. Test and then describe the effect on the force of gravity at the surface of a planet
resulting from each of the following: Return to the initial values for the Earth
after each test. i) Doubling the object mass. ii) Doubling the planet mass. iii)
Cutting the radius in half. iv) cutting the radius and the mass in half.
i) this doubles the force of gravity +1
ii) this also doubles the force of gravity +1
iii) this quadruples the force of gravity +1
iv) decreasing the radius increases the force of gravity by a factor of 4 but cutting
the mass in half decreases the force of gravity by a factor of 2 so the net increase
resulting from both changes is by a factor of 2.
3. If people were to make a colony on Mars, how much would an 80 kg astronaut
weigh on the surface of mars? By how much does this differ from his weight on
Earth?
The astronaut would weigh 298 N +1 as opposed to 784 N +1 on Earth. Multiply
g by 80 to get weight.
4. Jupiter is much larger than Earth and Mars. What would the same 80.0 kg
astronaut weigh on the surface of Jupiter (assuming there was a surface to stand
on)?
The same astronaut would weigh 1983 N +1 on Jupiter.
5. If the moon orbits the Earth at a distance of 3.84*108 m, what is the acceleration
due to gravity at this location? What is the acceleration due to gravity at half the
distance between the Earth and the Moon?
The acceleration due to gravity at that location is 0.00269 m/s2. Half the distance
between the Earth and the moon the gravitational acceleration is 0.0108 m/s2. To
do this simply find the position equal to half the moon’s orbital radius on in
column A and record g from column D.
6. “The heart of the galaxy known as M87 is a place of unimaginable violence. A
black hole up to seven billion times as massive as the Sun sits at the galaxy's
center -- one of the most massive black holes ever measured. As gas spirals into
the black hole, it's heated to millions of degrees, so it produces enormous amounts
of X-rays. Some of the hot gas around the black hole shoots back into the galaxy
in powerful jets that span thousands of light-years.” Using the data from the
Planetary Data Table for the super-massive black hole on the previous page,
determine the escape velocity from the assumed radius for the event horizon of
twice that of the orbit of Pluto. Based on this rough calculation, is the assumed
radius too high or too low? Hint: the speed of light is 3.0*108 m/sec.
Enter 1.39e40 in the mass cell and the given radius of 1.17e13 for the radius.
With an escape velocity of 3.99e8 this is too high meaning that the event horizon
would actually be larger than the orbital radius of Pluto. This implies that there
exists a black hole in our galaxy that is larger than our entire solar system.
7. Use goal seek to solve for the radius of the black hole in the previous problem if
the escape velocity can be assumed to be exactly equal to the speed of light.
What will the acceleration due to gravity be at the event horizon (the “surface”) in
this case?
The radius of the supermassive black hole would have to be 2.06e13 which would
be a little less than double the assumed radius. To do this set cell E6 to value 3e8
by changing cell D2. In this case the acceleration due to gravity would be 2179
m/s2.
8. How fast would an object have to orbit at a distance of 1012 meters from the event
horizon to remain in a circular orbit? Assume (ac = v2/r)
To calculate this simply change the central object radius to 2.16e13 and use g for
the surface. Ac = g = v2/r. as a result velocity will have to be 2.07e8 m/s or 2/3
the speed of light.
9. Can a Black Hole rip you apart? Let's see! (this will be hypothetical for several
obvious reasons) If you jump feet first into a black hole your feet will be
approximately 2 meters away from your head. With the data for the stellar black
hole on the spreadsheet (check escape velocity) enter an increment of 2 meters.
The difference in force will be the different forces that your head and feet will be
experiencing. What is the difference in the force of gravity at the event horizon
and 100 meters above it? Hint adjust the position increment to be 2.0 meters.
183,839,983,193 N will be the difference in force. This should be more than
enough to pull a person apart! 177,746,254,601 N is the force 100 m above the
surface. Even if you were 1 cm (0.01 m) tall the difference in force would still be
over 900,000,000 N and it would still be able to rip you apart.
10. Rigel A, a blue supergiant star, is part of the constellation Orion and is the
brightest star in the sky. On its surface, what is the escape velocity? Compare to
escape velocity on the surface of the sun which is 6.18*106m/s? What is the
reason for this difference?
The escape velocity of Rigel A is 3.23e5 +2 which is close to 20 times less than
that of the sun. The reason for this is due to the much larger radius and only
slightly larger mass. The reason for such a large radius and small mass is Rigel
A is a blue star that burns hotter and is therefore much less dense than our sun.
11. When a star runs out of fuel and slows its rate of energy emission, gravitational
forces can compress the star to the point that it can become a black hole. What
must the radius of Rigel A be to become a black hole? Hint: Escape velocity
should equal the speed of light, use goalseek to set escape velocity to equal 3e8 by
changing central object radius.
In order to become a black hole Rigel A would have to be compressed to have a
radius of 5.01e4 +1 for which it is much too hot. Gravity is not strong enough to
produce this effect yet. Over time as the star cools this can and will change for
countless stars destined to become stellar black holes.
12. If the Earth were to suddenly be compressed sufficiently to make it a black hole,
what would its radius have to be? What about our sun? Hint: See previous
problem hint.
Using the same approach as before:
Earth: r = 8.86e-3 +1or to about the size of a marble (1.7 cm in diameter)
Sun: r = 2.95e3 +1or a little less than 4 miles in diameter