Simple Nature - Light and Matter

38 Suppose an object has mass m, and moment of inertia I ofor rotation about some axis A passing through its center of mass.Prove that for an axis B, parallel to A and lying at a distance hfrom it, the object’s moment of inertia is given by I o + mh 2 . Thisis known as the parallel axis theorem.39 Let two sides of a triangle be given by the vectors A andB, with their tails at the origin, and let mass m be uniformly distributedon the interior of the triangle. (a) Show that the distanceof the triangle’s center of mass from the intersection of sides A andB is given by 1 3|A + B|.(b) Consider the quadrilateral with mass 2m, and vertices at theorigin, A, B, and A + B. Show that its moment of inertia, forrotation about an axis perpendicular to it and passing through itscenter of mass, is m 6 (A2 + B 2 ).(c) Show that the moment of inertia for rotation about an axis perpendicularto the plane of the original triangle, and passing throughits center of mass, is m 18 (A2 +B 2 −A·B). Hint: Combine the resultsof parts a and b with the result of problem 38.40 When we talk about rigid-body rotation, the concept ofa perfectly rigid body can only be an idealization. In reality, anyobject will compress, expand, or deform to some extent when subjectedto the strain of rotation. However, if we let it settle down fora while, perhaps it will reach a new equilibrium. As an example,suppose we fill a centrifuge tube with some compressible substancelike shaving cream or Wonder Bread. We can model the contents ofthe tube as a one-dimensional line of mass, extending from r = 0 tor = l. Once the rotation starts, we expect that the contents will bemost compressed near the “floor” of the tube at r = l; this is bothbecause the inward force required for circular motion increases withr for a fixed ω, and because the part at the floor has the greatestamount of material pressing “down” (actually outward) on it. Thelinear density dm/ dr, in units of kg/m, should therefore increase asa function of r. Suppose that we have dm/ dr = µe r/l , where µ is a√constant. Find the moment of inertia.41 When we release an object such as a bicycle wheel or a coinon an inclined plane, we can observe a variety of different behaviors.Characterize these behaviors empirically and try to list the physicalparameters that determine which behavior occurs. Try to form aconjecture about the behavior using simple closed-form expressions.Test your conjecture experimentally.Key to symbols:easy typical challenging difficult very difficult√An answer check is available at www.lightandmatter.com.Problems 295