Example #1: Inertia of disc with point masses

Practice: You build a wheel out of a thin circular hoop of mass 5 kg and radius 3 m, and two thin rods of mass 2 kg and 6 m in length, as shown below. Calculate the system’s moment of inertia about a central axis, perpendicular to the hoop.

Practice: A composite disc is built from a solid disc and a concentric, thick-walled hoop, as shown below. The inner disc (solid) has mass 4 kg and radius 2 m. The outer disc (thick-walled) has mass 5 kg, inner radius 2 m, and outer radius 3 m. Calculate the moment of inertia of this composite disc about a central axis perpendicular to the discs.

Practice: Three small objects, all of mass 1 kg, are arranged as an equilateral triangle of sides 3 m in length, as shown. The left-most object is on (0m, 0m). Calculate the moment of inertia of the system if it spins about the (a) X axis; (b) Y axis.

Consider a system composed of three thin rods each of mass m and length L that are welded together to form an equilateral triangle. What is the moment of inertia of this triangle for rotation about an axis that is perpendicular to the plane of the triangle and through one of vertices of the triangle? The moment of inertia of a rod rotated about its center of mass is Irod, cm =1/12mL2.
1. 17/12mL2
2.7/3mL2
3.5/6mL2
4.3/2mL2
5.1/2mL2
6.2/3mL2
7.11/12mL2
8. mL2

The figure below depicts a thin rod of length 2 m and having a mass of 1000 g with three small spheres attached that have a mass of 200 g each. What is the moment of inertia for this object rotating about an axis perpendicular to the rod at its end? (The middle sphere is at the center of the rod.)
A. 20 kg m2
B. 30/23 kg m2
C. 3√2 kg m2
D. 11/15 kg m2
E. 7/3 kg m2

The four masses shown in the figure are connected by massless, rigid rods. Assume that m exttip{m}{m}= 210 g.a) What is the x-coordinate of the center of mass?b) What is the y-coordinate of the center of mass?c) Find the moment of inertia about an axis that passes through mass A and is perpendicular to the plane of the image.

A diatomic molecule such as molecular nitrogen (N2) consists of two atoms each of mass M, whose nuclei are a distance d apart. What is the moment of inertia of the molecule about its center of mass?A. M d 2B. 2M d 2C. 4M d 2D. 1/2 M d 2 E. 1/4 M d 2

A sphere consists of a solid wooden ball of uniform density 800kg/m3 and radius 0.30 m and is covered with a thin coating of lead foil with area density 20kg/m2. Calculate the moment of inertia of this sphere about an axis passing through its center.

A wagon wheel is constructed as shown in the figure. The radius of the wheel is 0.300 m, and the rim has mass 1.41 kg. Each of the eight spokes, that lie along a diameter and are 0.300 m long, has mass 0.260 kg. What is the moment of inertia of the wheel about an axis through its center and perpendicular to the plane of the wheel?

A rotating object is formed by wrapping a cylinder with a thin plastic. The cylinder has a mass of 12 kg, a radius of 15 cm, and a height of 25 cm. The plastic wrap has a mass of 4 kg and is assumed to have a zero thickness. If the object rotates about the central axis of the cylinder, what is the moment of inertia of the object?

A thin, flat, uniform disk has mass M and radius R. A circular hole of radius R/4, centered at a point R/2 from the disks center, is then punched in the disk.a) Find the moment of inertia of the disk with the hole about an axis through the original center of the disk, perpendicular to the plane of the disk. (Hint: Find the moment of inertia of the piece punched from the disk.)b) Find the moment of inertia of the disk with the hole about an axis through the center of the hole, perpendicular to the plane of the disk.

A baton is made of a 10 cm rod with a mass of 500 g, with two 70 g masses attached to each end. What is the moment of inertia of the rod when it rotates about an axis, perpendicular to its length, halfway down the rod?

A thin uniform rod 50.0 cm long and with mass 0.320 kg is bent at its center into a V shape, with a 70.0 degree angle at its vertex. Find the moment of inertia of this V-shaped object about an axis perpendicular to the plane of the V at its vertex.

A 4.0-kg mass is placed at (3.0, 4.0) m, and a 6.0-kg mass is placed at (3.0, -4.0) m. What is the moment of inertia of this system of masses about the x-axis?
a. 160 kg•m2
b. 90 kg•m2
c. 250 kg•m2
d. 32 kg•m2

A uniform bar has two small balls glued to its ends. The bar is 2.00 m long and has mass 6.00 kg , while the balls each have mass 0.300 kg and can be treated as point masses. Find the moment of inertia of this combination about an axis:a) perpendicular to the bar through its centerb) perpendicular to the bar through one of the ballsc) parallel to the bar through both ballsd) parallel to the bar and 0.500 m from it