© Fernando Caracena, 1 November 2012
Uniform Circular Motion and Simple Harmonic Motion
Motion of objects in a circle and back-and-forth along a straight-line axis are two important, related topics in physics. Interest in the idea of circular motion originated in astronomy where circular motion is an approximation of planetary motions, the real orbits being ellipses of very small eccentricity. As the industrial age developed, the description of simple harmonic motion[see an excellent lecture and demo by Prof. Walter Lewin of MIT] as a component of uniform circular motion became a topic of interest: the motion of pistons moving back and forth is converted in the circular motion of wheels powered by drive shafts. Nowadays, the mathematical descriptions are part of the language of physics, which applies to the description of wave motion.
Formation of the Solar System
The model of the formation of the solar system is that 4.6 billion years ago a large rotating bubble of molecular hydrogen and dust, several light years across, contracted into a rotating flat disk that had a concentrated blob at the center, which formed into our present Sun. Because of common origins, and the ensuing dynamics of the solar system, the motion of the planetary orbits, and the spins of the planets and Sun are mostly aligned.
The orbits of most of the planets are ellipses of small eccentricity, almost circles, which are almost all contained in the same plane that passes through the center of the spinning Sun. The orbital angular momentum of a planet is a vector that points perpendicularly upward when you look down on its orbit, in which the planet travels in a counter clockwise sense. Further, the the planets spin in the same sense as Sun and
Fig. 1 A diagram of velocity vectors in uniform circular motion.
their axes of rotation are aligned with that of the Sun in the same sense as their own orbits; that is, their orbital angular momenta are aligned with their own rotational angular momenta (spin). Only Uranus is the exception. Some cosmic collision early in its history tilted its axis of rotation about 97.7 degrees off the axis of its orbit about the Sun. Below, we develop the mathematics needed to describe uniform circular motion as it applies to planetary motion. In talking about planetary motion, we need to develop the idea of angular velocity an angular momentum, which describe both the orbital motion of the planets about the Sun, and their spin about their own axis.
Angular Velocity and Centripetal Acceleration
We proceed by developing the subject from an intuitive level. I am assuming that the reader has some familiarity with geometry and algebra.
Consider an object in circular motion counter clockwise about a center (O, in Fig. 1) at a constant distance, r0, and constant speed, v. At a particular time, the velocity of the object is the vector v, which points to the left, perpendicular to the radius (Fig.1) along the tangent to the circle. When the radius rotates through an angle, Δθ, so does the tangent velocity, vector. The triangles formed by the position vectors r1 , r2 and r2 - r1 and velocity vectors v1 , v2 and v2 - v1 being similar triangles, we can state that
Δv/v=Δr/r0, (1)
where
Δv = |v2 - v1|
and
Δr = |r2 - r1| .
Dividing both sides by the time interval, Δt, gives the equation,
(Δv/Δt)/v=(Δr/Δt)/r0,
which gives the magnitude of the acceleration
a=v2/r0. (2a)
From Fig. 1, we can see that the direction of the acceleration is toward the center of the circle, viz., in the -r direction, which is defined by the unit vector, -r/r0. Therefore,
a=-r v2/r02. (2b)
Now the angular velocity of the object in motion, in radians (arc speed divided by the radius of the circular orbit) is
ω=v/r0, (2c)
in terms of which we can simplify (2b) to write it as
a= -ω2 r. (3)
Calculus Approach
This topic is developed rapidly and naturally by using vector calculus. We put the x-y plane in the plane of the planetary orbits, and have the planets' and Sun's spin axes pointing in the z-direction. The orbits of the planets are ellipses, but actually very close to circular orbits. Therefore, we use a mathematical description of planetary motion that approximates it as uniform circular motion. The following equation describes a position vector that can move in a circle at a fixed radius about the center and in the xy-plane:
r = r0 cos(θ) e1 +r0 sin(θ) e2, (4)
where θ is the angle that the position vector rotates counter-clockwise from the x-axis when looked at straight down toward the origin along the negative z-direction. Consider what happens when the only variable that depends on time in (1) is the angle of rotation, θ(t). In this case we apply a method of calculus called "the chain rule" to find the derivative of the position vector:
dcos(θ)/dt = dcos(θ)/dθ * dθ/dt
or
dcos(θ)/dt =-sin(θ)* dθ/dt . (5a)
Likewise, [show as an exercise that]
dsin(θ)/dt = cos(θ)* dθ/dt . (5b)
The last term on the RHS of (5a) and (5b) is the angular velocity of the planet in its orbit,
ω = dθ/dt . (5c)
Show (as an exercise) that using all of the above equations, we can solve for the orbital velocity as,
dr/dt = [- r0 sin(θ) e1 + r0cos(θ) e2 ] ω . (6a)
Notice that the following relation holds,
e3 x r =r0 [e3x e1 cos(θ)+e3x e2 sinθ)] , (7a)
or
e3 x r =r0 [e2 cos(θ) - e1 sinθ)] , (7b)
or
e3 x r =r0 [ - e1 sinθ) + e2 cos(θ)], (7c)
which can be used to rewrite (6a) as
dr/dt = ω e3 x r . (8a)
With further simplification, (8a) can be rewritten as.
dr/dt = ω x r , (8b)
or
v = ω x r , (8c)
where the vector ω is defined as
ω=ω e3 . (8d)
Note that when a planet orbits counter-clockwise in the xy-plane, the orbital angular velocity vector points up along the z-axis. The orbital angular velocity vector is perpendicular to the plane of orbital motion. Similarly, a spinning body, such as the Sun, has an angular velocity that points along the axis of spin, in this case, also in the positive z direction.
The acceleration of the orbiting body is computed by differentiating the velocity (8c)
a= dv/dt, (9a)
or
a= ω x dr/dt . (9b)
In this case, the angular velocity is a constant vector; were it a variable, another term would have had to be added to (9b) that contained the derivative of ω. We can simplify (9b) by substituting (8b) for dr/dt ,
a= ω x (ω x r), (9c)
which reduces to
a= ω (ω•r) - (ω•ω) r . (9d)
Looking at (4) and (8d) it is apparent that the first term on the RHS of (9d) vanishes because ω and r are at right angles to each other. As a result (d) is rewritten as
a= -ω2 r . (10)
Again, we find that the acceleration of an object moving in a circle at a constant angular velocity, is toward the center of the circle.
Connection with Gravity
The Sun holds the Earth in orbit around itself as a distance of about 1.50e11 to within a few percent owing to the eccentricity of the orbit. The force of gravity, applying Newtons law of gravitation, exerted by the Sun on the Earth is
-G Ms me ur/r02 = -me ω2 r , (11a)
or
-G Ms /r02 = - ω2 r0 . (11b)
When (11b) is solved for Ms, we get the following equation:
Ms = ω2 r03 /G
A computation with python code (below) yields the mass of the Sun, based on the time it takes the Earth to orbit once around the Sun (T=1 year = 3.16 x 107sec). The angular velocity is given by
ω = 2π/T
Also note that the gravitational constant has the value,
G=6.67300e-11 m3 kg-1 s-2
and that the distance between the Sun and the Earth is
r0=1.50 x 1011 m.
The mass of the Sun, computed as described above, turns out to be
Ms = 1.99 x 1030 ±0.05 x 1030 kg
an error resulting from the variability of the distance, r0 .
According to Google the mass of the sun is 1.9891 × 1030 kg, so that we see that our calculation agrees pretty well, that is , its range of values due to error (1.94-2.04) x 1030 bracket the value 1.9891 × 1030 kg.
Note that python will give much higher precision in its answers than what is called for. The person doing the calculations must realize that the numbers used have some uncertainty related to them, and therefore, the answer given by python must be rounded off to the proper, significant figure, and the error range given.
Conclusion
Here we have shown how it is possible to determine the mass of astronomical bodies by observing the orbits of their satellites. We have used a model of uniform circular motion for the orbit of planet Earth about the Sun to estimate the mass of the Sun. The principle source of error for this calculation is that the Earth's orbit is an ellipse not a circle, but the departures from the circular orbit are small (about 2.4%). We could reduce the error by using an elliptical orbit in the calculations; but, that extra complexity would be at the cost of simplicity. Our main target in this discussion is what we shall discuss in the future, angular momentum, which is another important variable in physics. However, before discussing that topic, we should discuss another topic similar to this one, namely, simple harmonic motion.
python code
ipython --pylab
from pylab import *
r=1.50e11 #meters +/-0.03e11
T=24.*3600.*365.25
G=6.67300e-11 #m^3 kg^-1 s^-2
w=2.*pi/T
Ms=w**2*r**3/G #kg
asn=r*w**2
print Ms