Physics 150 Fall 2003
Solution to HW 2
1. There can be many answers. Possible ideas are:
a. One observes the sun, moon, and stars moving but do not sense that Earth is moving
b. There is really no direct evidence. (Later Aristarchus
of Samos measured the size of the sun to be much
larger than that of the Earth suggesting its greater importance in the heavens
than that of the Earth.) But it could be argued that the sun is more powerful
in some way and therefore at the center.
Indeed the sun
god was the supreme god in
c. The known retrograde motion of certain “stars” (planets) and the varying orbits of the Sun and Moon might suggest that there is no center. This would appear to be a more philosophical argument.
Question C: Galileo invoked two principles in order to show that the Earth could be moving without us knowing it. The first was inertia: he knew when he rolled a ball off of a table that, after it left his hand, it continues to move at the same speed “away from him” (in the horizontal direction) throughout its entire path, even while it is falling. That meant that if a person jumps, even though the Earth is moving, the jumper will not notice the motion of the Earth under herself because she is moving at the same speed with the Earth. The second principle was superposition: the ball that he rolled off the table continued to roll at the same speed (in the horizontal direction) even after the ball starts falling. Gravity acts in the vertical direction and thus doesn’t affect the horizontal motion of the ball. Therefore, the two motions (horizontal and vertical) are independent and one can “superimpose” the two independent motions in order to observe the real motion of the ball. This discovery implied that gravity would act exactly the same whether the Earth was still or in motion.
Note: Show all work on numerical problems. Do not write a numerical answer only.
Chapter 4, Question B:
To answer this question, we need to find information available at the time that would give the ratio of the mass of the sun to the mass of the earth. Two pieces of information can be used to get information on the mass of the earth:
1) the acceleration of gravity ( F = GmMe/Re2 = mg or g = GMe/R2 or GMe = g Re2, where Me = mass of the earth and Re = radius of the earth).
2) The period of the moon is given by F = GmM/Rem2 = ma
= m v2/Rem
= m (2pi Rem /Pm)2/Rem or GMe = (2pi)2 Rem3 /Pm2 where Rem3 is the earth-moon distance and Pm is the period of the moon’s orbit (28 days).
Similarly, to point 2) above the mass of the sun is given by GMs = (2pi)2 Res3 /Pe2 where Res3 is the earth-sun distance and Pe is the period of the earth’s motion (365 days)
Thus the ratio GMs /GMe = Ms /Me = (Res3 /Pe2 )/(Rem3 /Pm2) where all the quantities on the right are known. A formula can also be found in terms of g.
Homework problem #3
The picture drawn shows a right triangle. This means that we can use trigonometry to calculate the distance from the Earth to the moon, d. This distance is the hypotenuse of the right triangle. The Earth’s radius r is the adjacent side to the angle L. Using the trick: SOH-CAH-TOA (sine equals opposite over hypotenuse – cosine equals adjacent over hypotenuse – tangent equals opposite over adjacent), we see that we want to use the cosine of L. cos L = adj/hyp = r/d or d = r/cos L. We use the approximate radius of the Earth to be r = 3900 miles then d = 3900/cos(89.1) = 248,300 miles.
Homework problem #4
We now have two right triangles. If we isolate one triangle then it has a small angle of 0.5*A=1/4 degree. The hypotenuse is again the distance from the Earth to the moon (subtracted by the Earth’s radius this time except because the distance to the moon is much larger than the Earth’s radius we will just use the value from problem 1) Using trig again we see that sin(0.5*A) = opp/hyp = moon’s radius/distance Earth to moon. Or moon’s radius = 238,000 miles*sin(0.25) = 1040 miles.
Homework problem #5
There are three components to the accuracy of this measurement. 1) the accuracy in knowing the Earth’s radius 2) the accuracy in knowing the values of trigonometric functions 3) the accuracy in measuring the angles above.
Errors in the Earth’s radius are affected by the Earth not being perfectly spherical. This kind of error is not too serious because our answer depends linearly on its measurement, e.g. a 1% error in the Earth’s radius causes a 1% error in the distance. For example, let’s say we used 3939 miles for the Earth’s radius (1% difference from the value we used) then our answer would have come out as around 251,000 miles which is roughly a 1% change from our original answer.
The second type of error is also not too serious. For example, my calculator shows that the value of cos(89.1) = 0.01570. The Ancients would not have had this convenience and would instead have used tables constructed by physically measuring ratios from triangles. If we imagine a 1% error in this valu, our answer is still within 1% of our original answer.
However, the third type of error can be disastrous. For example, if we measured the angle to be 89.95 degrees rather than 89.1, which is less than 1% difference in the angle, then our answer would have been 4.77 million miles which is a 1900% change.