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Practice: Gravitation
1. Everyday Gravitational Acceleration
Two blocks of mass 𝑚! and 𝑚! are placed 1.0 𝑚 apart. The coefficient of static
friction for 𝑚! is 0.40. Assume 𝑚! is fixed in place. What minimum mass must
mass 2 have in order that mass 1 move towards it due to Newton's law of
gravitation?
Given: 𝐺 = 6.67×10!!!
!!!
!!!
Solution:
As mass 2 gets larger and larger, the gravitational force grows. The frictional force however manages to
hold it in place exerting an equal but opposite force, to a limit. We need to know what the maximum force
is that static friction can exert on mass 1 and equate it to the gravitational force. We solve for when
2. The Mass of the Sun
Estimate the mass of the sun given the Earth's distance from the sun 𝑟 =
1.50×10!! 𝑚.
Assume the Earth follows a circular orbit instead of an elliptical one.
Given: 𝐺 = 6.67×10!!!
!!!
!!!
Solution:
The attractive force of gravity acts as a centripital force.
We know r and the gravitiational constant G, but we still need to find
, the angular velocity of the Earth.
Angular veloctiy is defined as anglular displacement divided by time, and we know how long it takes the
Earth to revolve around the sun (1 year), so...
It takes 365 days for the Earth to go around the sun once. We need to change this to seconds so as to
get SI units.
The actual mass of the sun is
, less than 1% error!
3. Putting a Satellite into geostationary orbit
Here’s a picture of Mariner 4:
If you want to put a satellite into a geostationary orbit (Mariner 4 never actually
was), meaning that it stays above the same position over the Earth at all times, at
what altitude above sea level must you have it orbiting?
Given: Mass of the Earth: 𝑚! = 5.975×10!" 𝑘𝑔, Radius of the Earth: 𝑟! =
6.371×10! 𝑚
Solution: If the satellite is to remain over the same point on the Earth, the satellite will have to match the
rotation of the Earth on its own axis. This means that the angular velocity of the satellite must equal that
of the Earth.
The Earth rotates once around its axis in 24 hours. We need to convert this to radians per second.
The satellite will be held in orbit by the gravitational attraction of the Earth, but we need to find the altitude
that the gravitational force provided equals the centripital force necessary to keep the satellite orbiting
and not falling into the Earth at the angular velocity we found above. Therefore we set the two equations,
the gravitational force and the centripital force, equal to each other and solve for r.
We have to remember to subtract off the radius of the Earth to get the altitude over the surface of the
Earth.
4. Comparing Gravitational Accelerations
a. Given the mass of the moon of 7.35×10!! 𝑘𝑔 and the radius of the moon
as 1.74×10! 𝑚, what is the gravitational acceleration on the surface of the
moon?
Solution: We have two equations that will allow us to calculate g of the moon without having to use the
mass of an object.
Both of these equations have m, the mass of the object, but we can set them equal to each other and
cancel m out.
Now we have an equation to find g which is the acceleration due to gravity.
b. A planet has three times the mass of Earth with a radius twice as large as
our own. What fraction of Earth's gravitational acceleration would a mass
feel on the surface of this planet?
Solution: We go through the same steps as from part a. and get the expression for solving for g
Now we plug in 3M for the mass and 2r for the radius to see how it compares to the earth's g.
The g of this planet would be 3/4 of g on Earth.
5. The Distance from the Earth to the Moon
The orbital period (T) of the Moon around the Earth is 29.53 days. Calculate the
radius of orbit of the Moon assuming the orbit is circular. You are given:
Given: Universal Gravitational constant: 𝐺 = 6.67×10!!!
𝑚! = 5.98×10!" 𝑘𝑔
!!!
!!!
, Mass of the Earth:
Solution: At first, let us look for the forces acting on the moon. The Moon rotates around the Earth, so
centrifugal force is exerted radially outward of the orbit. Gravitational force is also exerted radially inward
of the orbit. That is, the centrifugal force and gravitational force point opposite to each other. Since there
is no radial acceleration, these forces must be equal in magnitude. Then, we get the following equations:
We can calculate the speed of the Moon.
The speed is also the same as the orbit radius times angular velocity.
This answer is a rough estimation of the radius. The actual mean distance to the Moon from the Earth is
384,400km.
6. Imagine a solid sphere of uniform density 𝜌 with radius r. From this sphere, take
out a smaller sphere radius r/2 so the original sphere is left with a spherical hole
in the location visible in the picture. Place the center of a sphere with the same
density 𝜌 but radius r/2 a distance 4r away from the center of the sphere with the
hole in it. What force do these two objects feel due to gravitational effects?
Solution: Since there is a hole in the larger mass, we can calculate the attractive force between the two
objects if there were no hole and then subtract off the attractive force between the smaller mass and a
mass the size of the hole the same distance away.
To find the mass given the density, we need to multiply the density times the volume of the sphere.
Now that we know the attractive force that the spheres would have if there weren't a hole, we see how
much attractive force the hole is responsible for.
Now we can just subtract off the force that the hole is
responsible for from the solid sphere