Newton's Laws of Motion and Gravitation 43EXERCISE 2.1.2r Let point mass m move in a planar path C given by r(t),
where r is the position vector with origin O.
(a) Use formulas (1.3.24), Pa9^ 35, and (2.1.4) to express the angular mo-
mentum of the point mass about O in the coordinate system {r,6).
(b) Suppose that the point mass moves in a circular path C with radius R
and center O. Denote the angular speed by w(i). (i) Compute the angular
momentum LQ of the point mass about O and the corresponding torque,
(ii) Find the force F that is required to produce this motion, assuming the
frame O is inertial. (Hi) Write F as a linear combination ofr and 6. (iv)
How will the expressions simplify ifui(t) does not depend on time?
As we saw in Exercise 2.1.1, the Second Law of Newton implies the
First Law. For further discussion of the logic of Newton's Laws see the
book Foundations of Physics by H. Margenau and R. Lindsay, 1957. Re-
garding the First Law, they quote A. S. Eddington's remark from his book
Nature of the Physical World, first published in the 1920s, that the law,
in effect, says that "every particle continues in its state of rest or uniform
motion in a straight line, except insofar as it doesn't." This is a somewhat
facetious commentary on the logical circularity of Newton's original formu-
lation, which depends on the notion of zero force acting, which can only
be observed in terms of the motion being at constant velocity. The same
logical difficulty arises in the definition of an inertial frame as a frame
in which the three laws of Newton hold. We do not concentrate on these
questions here and simply assume that the primary inertial frame, that
is, a frame attached to far-away, and approximately fixed, stars is a good
approximation of an inertial frame for all motions in the vicinity of the
Earth. The idea of this frame goes back to the Irish bishop and philoso-
pher G. Berkeley. The deep question "What is a force?" is also beyond the
scope of our presentation; for the discussion of this question, see the above-
mentioned book Foundations of Physics by H. Margenau and R. Lindsay,
or else take as given that there are four basic kinds of forces: gravitational,
electromagnetic, strong nuclear, and weak nuclear. In inertial frames, all
other forces result from these four.
Newton discovered the Law of Universal Gravitation by combining his
laws of motion with Kepler's Laws of Planetary Motion. The history behind
this discovery is a lot more complex than the familiar legend about the apple
falling from the tree and hitting Newton on the head. As many similar
stories, this "apple incident" is questioned by modern historians. Below,
we present some of the highlights of the actual development.