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.

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Sections | |||
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Intro to Moment of Inertia | 30 mins | 0 completed | Learn |

Moment of Inertia via Integration | 19 mins | 0 completed | Learn |

More Conservation of Energy Problems | 55 mins | 0 completed | Learn |

Moment of Inertia of Systems | 23 mins | 0 completed | Learn |

Conservation of Energy in Rolling Motion | 45 mins | 0 completed | Learn |

Moment of Inertia & Mass Distribution | 10 mins | 0 completed | Learn |

Intro to Rotational Kinetic Energy | 17 mins | 0 completed | Learn |

Energy of Rolling Motion | 18 mins | 0 completed | Learn |

Types of Motion & Energy | 24 mins | 0 completed | Learn |

Parallel Axis Theorem | 14 mins | 0 completed | Learn |

Conservation of Energy with Rotation | 36 mins | 0 completed | Learn |

Torque with Kinematic Equations | 59 mins | 0 completed | Learn |

Rotational Dynamics with Two Motions | 51 mins | 0 completed | Learn |

Rotational Dynamics of Rolling Motion | 27 mins | 0 completed | Learn |

Example #1: Inertia of disc with point masses

**Transcript**

Hey guys. So, let's check out another example of a moment of inertia problem. So, here we have a solid disk. Remember, a solid disk has the same moment of inertia as a solid cylinder, and when I tell you the shape I'm telling you which equation to use for I. So, I of the disk is 1/2 m, r squared, I'm also given that it has a radius of 4. So, R equals 4 and a mass of 10, okay? And then we're going to add two small objects on top of it, the fact that it's saying two small objects and it's not giving you a shape for the object it's an indication that these are going to be treated as point masses, okay? And these are the two objects here. So, I'm going to call this one so this is m1 and this is m2, okay? It says, the object on the left has a mass of two kilograms and it's placed halfway between the disks centers, the center disk is here, in the edge. Now, the distance between the center and the edge is the radius, right? Which is 4, if you are halfway between the center and the edge you, your distance is half the radius, okay? So, this distance here is half the radius. So, I'm going to call this R1 because is the distance for mass 1 and it is 1/2 of the radius, which is 2, and the other guy, the other object is 3 kilograms in mass, 3 kilograms, and it's placed at the edge of the disk. So, if you're all the way at the edge your distance, let's call this R2, is the same as the radius, which is 4, okay? So, I'm giving you all the information need to calculate the system's moment of inertia. Now, system is a combination of the disk with the masses. So, I system is I1 plus I2, right? Object 1 object 2, plus I disk and remember for every one of these you have to determine is this at point mass or is this a shape? Well, one and two are point masses, we talked about this here. So, I'm going to write m1, R1 squared plus m2, R2 squared and the disk is a shape and it has moment of inertia given by 1/2 mr squared, 1/2 Big M big R squared. Now, all we got to do is plug in the numbers very straightforward. So, what I'm going to do is plug the masses the masses are 1, 2 and 10. So, I'm sorry, 2, 3 and 10, 2, 3 and 10. So, I'm going to do 2, 3 plus half of 10R squared, okay? And then all we have to do is plug in the r's, this R here is the radius it's very straightforward, we know that the radius is a four. So, I'm going to put a 4 here.

Now, these r's, we have to slow down for a little bit, these are the distances between the center the axis of rotation, which is in the center, and what the object is, little r is the distance between the object and the center and we already have these figured out here, it's 2 and 4 so the 2 kilogram has a 2 meter distance and the 3 kilogram has a 4 meter distance, okay? So, let's just do this real quick, this is going to be 8, this is going to be 3 times 16. So, that's 48 and this is going to be 80, okay? So, we have 8 plus 48 plus 80 and this is going to be 136, 136 kilograms meter squared, cool? That's it for this one, hopefully got it, let me know if you have any questions.

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.

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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

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