Physics 111 HW 22
Assigned 14 November 2012
I01. Consider the situation illustrated at right. Four point masses (each
1kg) are held together with stiff rods of negligible mass, forming four
corners of a rectangle. Find the moment of inertia about the three axes
shown (dotted lines).
0.4 m
0.2 m
I02. A thin rectangular plate of width a and height b has a mass M and is
mounted on an axis that is parallel to its height as seen in the figure.
a) Show that the moment of inertia of this plate is (1/3)Ma2.
b) Find the moment of inertia of this plate about an axis that lies in the
plane of the plate, passes through the center of the plate, and is parallel
to the axis shown in the figure. (Use the parallel axis theorem for parts
b and c.)
c) Find the moment of inertia of this plate about an axis that lies in the
plane of the plate, passes through the center of the plate, and is
perpendicular to the axis in part a.
b
a
I03. Consider a long, thin rod of length L and mass M. Calculate the moment of inertia of this
rod about an axis that is perpendicular to the rod and a distance h from one of its ends by
using two methods:
a) Starting with the formula for I for an axis that is perpendicular to the rod but goes through
its center, use the parallel axis theorem.
b) Use integration techniques.
I04. A uniform, very thin bar has two small balls glued to its ends. The bar is 2.00 m long with
mass 4.00 kg, while the balls each have mass 0.500 kg and can be treated like point masses.
Find the moment of inertia of this combination about each of the following axes:
a) an axis perpendicular to the bar through its center;
b) an axis perpendicular to the bar through one of the balls;
c) an axis parallel to the bar through both balls;
0.5 kg
0.5 kg
4.0 kg
d) an axis parallel to the bar and 0.500 m from it.
2.00 m
I05. The three uniform objects shown in the diagram below have the same mass m. Object A is
a solid cylinder with radius R. Object B is a hollow, thin cylinder with radius R. Object C is
a solid cube whose sides are of length 2R. The axis of rotation of each object is
perpendicular to the page and through the object’s center of mass.
a) Which object has the smallest moment
of inertia? Explain.
b) Which object has the largest moment
of inertia? Explain.
c) Where would the moment of inertia of
a uniform solid sphere of mass m and
radius R rank (axis through the center of mass)? Explain.
I06. A thin rod of length 2m has a varying mass density given by λ = Ax3, where A = 5.3 kg/m4
is a constant and x is the distance from the left end of the rod.
a) What is the total mass of the rod?
b) How far from the left end of the rod is the center of mass of the rod?
c) Use integration techniques to determine the moment of inertia of the rod about an axis
through its left end and perpendicular to the rod.
d) Use the parallel axis theorem to determine the moment of inertia of the rod about
i) an axis perpendicular to the rod and through its center of mass, and
ii) an axis perpendicular to the rod and through its right end. (This turns out to be less
than that calculated from part c). Does that make sense?)