Simple Nature - Light and Matter

The parallel axis theorem example 16⊲ Generalizing the previous example, suppose we pick any axisparallel to axis 1, but offset from it by a distance h. Part (2) ofthe previous example then corresponds to the special case ofh = −1.0 m (negative being to the left). What is the moment ofinertia about this new axis?⊲ The big ball’s distance from the new axis is (1.0 m)+h, and thesmall one’s is (2.0 m)-h. The new moment of inertia isI = (2.0 kg)[(1.0 m)+h] 2 + (1.0 kg)[(2.0 m) − h] 2= 6.0 kg·m 2 + (4.0 kg·m)h − (4.0 kg·m)h + (3.0 kg)h 2 .The constant term is the same as the moment of inertia about thecenter-of-mass axis, the first-order terms cancel out, and the thirdterm is just the total mass multiplied by h 2 . The interested readerwill have no difficulty in generalizing this to any set of particles(problem 38, p. 294), resulting in the parallel axis theorem: If anobject of total mass M rotates about a line at a distance h fromits center of mass, then its moment of inertia equals I cm + Mh 2 ,where I cm is the moment of inertia for rotation about a parallel linethrough the center of mass.Scaling of the moment of inertia example 17⊲ (1) Suppose two objects have the same mass and the sameshape, but one is less dense, and larger by a factor k. How dotheir moments of inertia compare?(2) What if the densities are equal rather than the masses?⊲ (1) This is like increasing all the distances between atoms by afactor k. All the r’s become greater by this factor, so the momentof inertia is increased by a factor of k 2 .(2) This introduces an increase in mass by a factor of k 3 , so themoment of inertia of the bigger object is greater by a factor ofk 5 .4.2.4 Iterated integralsIn various places in this book, starting with subsection 4.2.5,we’ll come across integrals stuck inside other integrals. These areknown as iterated integrals, or double integrals, triple integrals, etc.Similar concepts crop up all the time even when you’re not doingcalculus, so let’s start by imagining such an example. Suppose youwant to count how many squares there are on a chess board, and youdon’t know how to multiply eight times eight. You could start fromthe upper left, count eight squares across, then continue with thesecond row, and so on, until you how counted every square, givingthe result of 64. In slightly more formal mathematical language,we could write the following recipe: for each row, r, from 1 to 8,consider the columns, c, from 1 to 8, and add one to the count for270 Chapter 4 Conservation of Angular Momentum