Demonstrating Circular Motion with a Model Satellite/Earth System
A number of interesting demonstrations of circular and satellite motion have been
described in this journal. 1-3 This note presents a variation of a Centripetal Force
apparatus found in G.D. Freier and F.J. Andersons’ A Demonstration Handbook for
Physics4 which has been modified in order to demonstrate both centripetal force and
satellite motion. Nice discussions of satellite motion may be found in a number of
textbooks.5-7 Here I describe how to construct the apparatus quite simply using readily
available materials.
Making the Model Satellite/Earth System
The major items required are glass tubing, an Estes or other model rocket, nylon
upholstery thread, a support stand, wooden sphere and hooked masses. As shown in fig.
1, the glass tubing should be fire polished on both ends and mounted, slightly off-center
in a wooden sphere/globe. The sphere/globe is mounted at the top of a support stand.
Upholstery thread is attached to the center of gravity of a small rocket, and threaded
through the glass tube. The other end of the thread is then attached to a paper clip (bent
into the shape of a hook) on which various small hooked masses can be hung. To
“launch” the satellite, the rocket is drawn away from the globe and tossed tangentially in
the horizontal plane. The rocket will then “orbit” the globe(Earth) for approximately 30
seconds until returning to “Earth”.
Demonstrating Angular Speed and Centripetal Force
Various demonstrations involving circular motion can be performed with this
apparatus. Angular speed can be measured by simply counting and timing rotations. Also,
centripetal force can be demonstrated and explained by first comparing the weight
hanging on the hook to the weight of the rocket:
(1.1)
W1 W2
where W1 is the downward weight of the rocket, and W2 is the much larger weight on the
hook. Once this is established, the rocket can then be sent into “orbit,’ in which case the
added horizontal force due to centrifugal action, Fc, plus the weight of the rocket
balances the weight on the hook:
[(Fc)2 + (W1)2]1/2 = W2.
(2)
Since Fc is related to the mass, orbital radius and tangential velocity of the rocket, W1 is
related to the mass and gravitational acceleration of the rocket, and W2 is related to the
mass and gravitational acceleration of the hooked mass, we can rewrite Eq. (2) as
[(m1 v2/r)2 + (m1g)2]1/2 = m2g
(3)
Where m1 is the mass of the rocket, v is the tangential orbital velocity of the rocket, r is
the orbit radius, and m2 is the mass on the hook.
To verify this relationship for the model, an experiment was performed in which v was
obtained by measuring the radius of the rocket’s orbit with a meter stick, calculating the
orbital circumference, 2r, and dividing that by the orbital time period, which was
measured with a digital stopwatch. Velocity data from six trials were averaged and
applied to equation 3. From the data it was found that the left-hand side of equation 3
agreed with the right hand side of equation 3 within 2.2 %. Given that the time period
was measured by hand, measurement error likely accounted for the difference between
the two values.
Demonstrating Kepler’s Third Law
During the rocket’s flight, the orbital radius decreases gradually until the rocket comes to
rest back on “Earth”. Kepler’s third law, which can be expressed as
T = 2r3/2/(GME)1/2
(4)
Shows that the period of a satellite in circular orbit is proportional to the three-halves
power of the orbital radius. To verify that this relationship holds true for the model, an
experiment was performed in which the orbital time period was measured, using a digital
stopwatch, at 5 different radii. The curve of best fit of time period versus radius, shown in
figure 2 below, suggests the model closely approximates the relationship in Kepler’s third
law.
Figure 1. Satellite, globe, tube, thread and stand.
References
1. John L. Makous, “Variations on a circular motion lab”, Phys. Teach. 38, 354-355
(Sept. 2000).
2. Bill Jameson, “Additions to a circular motion lab”, Phys. Teach. 37, 545-546
(Dec. 1999).
3. Knip, et. al., “Simulating realistic satellite orbits in the undergraduate classroom”
Phys. Teach. 43, 452-455. (Oct. 2005)
4. G. D. Freier, F.J. Anderson, A Demonstration Handbook for Physics (American
Association of Physics Teachers, Stony Brook, NY, 1981).
5. John D. Cutnell, Kenneth W. Johnson, Physics (John Wiley & Sons, Inc.,
Hoboken, NJ, 2004).
6. Paul W. Zitzewitz, Physics: Principles and Problems Glencoe/McGraw-Hill,
Columbus, OH, 2002).
7. David Halliday, Robert Resnick, Jearl Walker, Fundamentals of Physics (John
Wiley & Sons, Inc., Hoboken, NJ, 2004).