Gravitational Force and Orbits
When we know the motion of an object, for example a circular orbit, we can determine the force
required to make the object move in a circle. If that force is gravitational then we can determine
the mass of the thing that is pulling the object (for example the sun pulling on the Earth, or the
Earth pulling on the moon). This is handy since we cannot bring the sun into the lab and measure
its mass.
A)
Verify circular motion and centripetal acceleration.
When objects move in a circle their (today a tennis ball) velocity changes direction, but the speed
does not change. This is an acceleration called centripetal acceleration, ac.
Eq. 1
If we know the acceleration of an object by watching it move in a circle, then we also know the
force that must make that happen. Newton’s second law relates the force.
Eq. 2
You will take an object (a tennis ball) and spin it with known radius and also measure the time it
takes to move in a circle. You will determine the centripetal acceleration given above using eq1
and multiply by the mass. This gives you the right hand side of Newton’s second law. We expect
the number on the left side to be the same. You will hang a mass from a string tied to the tennis
ball (0.500kg hanging for now). This, with some help from gravity provides a pulling force on
the string. The force due to the pulling on the string here is simply (mhangingg)=4.9N. See the
picture below to figure out what the set up will be.
You will spin the tennis ball with the string sliding through a tube. Spin it until one of the marker
flags hits the bottom of the tube. This tells you the pre-measured radius (marked). Your hanging
mass is a bottle of water filled to the line for 0.500kg. You will need to be coordinated to do this.
Spin the ball so that the marker REMAINS at the bottom of the tube. Your timer/counter person
will time and count 20 full revolutions. You should repeat this several times and take an average.
Once you have your average you will still need to divide by 20 to get the time for one revolution.
You should do this in the room, atrium, or outside where there is clearance. Determine the speed
of the ball and the centripetal acceleration using Eq. 1. After you have your data for Time,
measure the mass of the ball on a scale, record the time for one rev, and calculate both v and then
a.
mball=
T0.705m=
R=0.705m.
vmeas=____________
ameas=___________
Now you can do this again with a longer radius (bigger circle).
T1.41m=
vmeas=____________
R=1.41m.
ameas=___________
You have the mass of the ball and know that you hung 4.9N at the end of the string. Check to see
that you get approximately 4.9N for each “orbit” when you calculate the centripetal acceleration
times the mass (which should equal the hanging force).
Centripetal force = mballa=
Centripetal force =mballa=
Do your results verify that centripetal acceleration gives the correct force?
(If not, return and try again).
B)
Determine the mass of the Earth (do prior to lab)
Now that we can believe our expression for centripetal force, we will investigate other cases
where we know the motion of an object. Rather than a string making objects move in circles we
can investigate the nearly circular orbit of the moon about the Earth. The real orbit is elliptical
but we will use a circle as a good approximation. The distance from the Earth to the Moon is
3.844x105km and the orbital period is 27.322days. Determine the speed of the moons orbit about
the Earth (in meters per second).
vmoon=
It is gravity rather than a string pulling on the moon. So we will use Newton’s universal law of
gravitation on the left side of Newton’s force law, and we will use centripetal force (from the
observed motion mmoon v2/r) on the right side. I’ll write it out for you.
Eq. 3
You are going to do a little math here and determine the mass of the Earth. DO THIS IN
SYMBOLS BEFORE PLUGGING ANY NUMBERS IN. Then plug the numbers in. Look up
and check your numbers against the known mass of the Earth in your book.
MEarth=
C)
Keplers Law
Using the same device, we can model gravitational force and verify Kepler’s third law. By now
you are aware that you must be careful spinning and timing the circular motion of the tennis ball.
You will make measurements of the time period for two separate cases very similar to Part I.
However this time we will carefully make the string pull on the tennis ball in a way that mimic’s
gravitational force.
F
Gm1m 2
r2
Eq. 4
This tells us how gravitational force changes if we change the orbital radius. For example if a
planet were twice as far from the sun, then the gravitational force would change to one fourth of
the original value. We can change the force on our spinning tennis ball by changing the amount
of water in the hanging mass. We are making the string MIMIC Gravity. Numbers that are easy
to use in lab have been provided. You will have one trial where you hang 0.500 kg mass from the
string, and measure the time period with the radius set at 1.00m. Then you will reduce the mass
to 0.250kg. This changes the force by a factor of two (reduction). For gravitational force this
change corresponds to moving 1.41 times further from the object you are orbiting. So with your
0.250kg mass you will use a radius of 1.41m in order to MIMIC Gravity. This information is
summarized below. Again you should take several trial runs to yield good measurements for T.
Run 1:
R1=1.00m
mass hanging=0.500kg
T1=
Run 2
R2=1.41m
mass hanging=0.250kg
T2=
Now analyze results using Keplers third law (this law can be derived directly from Newton’s
Universal Gravitation).
T2=kR3
Eq. 5
For each run calculate Kepler’s constant using your data. You should be obtaining the same
number for each, though you will see minor differences since measurements are not perfect. You
may keep units you used to make measurements in lab, but you must have units.
k(run 1)=
k(run 2)=
%difference between k’s =difference/average *100%
Discuss briefly any reasons you observe in the experiment to explain why the numbers are not
exactly the same. Think about measurements you have made.
Once you have your k (average), and have confirmed that both are “the same” within
experimental uncertainty, you can ask---what gravitational source (mass) would I need to have to
replace my string? Recall that we made the string mimic gravity by selecting how to pull on the
string.
Use your “k” or your data for R and T, to determine what mass an object would need to have in
order to pull on the tennis ball with the same force as the string was made to. You can use
Newton’s law of gravitation and centripetal acceleration to write a statement like eq. 5 in terms of
2
GM  2r  1

constants:
Which you should rearrange to look like equation 5. Then use

r2
 T  r
your average measured k to determine the mass.
M=
Can you explain why we used a string to pull on the tennis ball rather than using some object to
pull gravitationally?