Problem 1. A moment of inertia. Figure 1 shows a rigid...

Problem 1. A moment of inertia. Figure 1 shows a rigid equilateral triangular frame in the xy plane. The frame is made of three thin rods, each with length L and mass m, and one corner is located at the origin O as shown. The total mass of the frame is 3m. The x-axis is a symmetry axis of the frame. Figure 1. A rigid equilateral triangular frame is made of three thin rods, each with length L and mass m, for a total mass 3m. The x axis is a symmetry axis of the frame. a) Calculate the moment of inertia of the triangle for rotation about the x axis. b) Calculate the moment of inertia of the triangle for rotation about the y axis. c) Write the angular momentum of the triangle rotating with angular velocity ? about the y axis in unit vector notation, using only the variables m and L and required numbers and unit vectors. Note: there are strictly speaking two correct answers depending on the sense of rotation. Write one of the correct solutions for full marks.

Transcript

00:01 In this question there is a xy plane this is x plane and this is y plane the equilitary triangle this is equal triangle the length of the equal triangle is l and the mass is m this is made from three rods so one rod mass is capital m and one rod length is l so we can write here masses and length so this is the origin this is the origin and total mass of the frame is three times m so this is symmetric about x -axis so we are calculating the moment of inertia of the triangle about x -axis and moment of inertia of this triangle about y -axis finally we are calculating the angular momentum about y -axis so in first part we are calculating the moment of inertia about x -axis so this is first rod, second rod, and this is third rod.

01:06 So let's find out the moment of inertia.

01:08 This is sum of all the rods.

01:11 So you can see this is symmetric about x -axis.

01:14 So we are considering this length, this is y, and this distance is x.

01:21 This angle is 30 degree and this angle is 30 degree because this is equilateral triangle.

01:26 So equal to triangle is 60 degree and it is symmetric, so that's why it is half.

01:30 The value of y is equal to x this is sign 30 degree so this is small movement of inertia for rod one this is equal to y square times small mass we are considering small mass dm now and the thickness is d x so we can write the mass per unit length dm divided by d x is equal to mass divided by l so you will get the value of dm that is m divided by l times d x so if you integrate this is moment of inertia or rod 1 y square integration this is mass time divided by length times d x so we are integrating from 0 to l length so i1 is equal to this is 0 to l this is x sine 30 degrees so this is x divided by 2 square m divided by l is constant and this is d x so you so after solving integration this is x cube divided by three times four because two square is four and this is zero to l m divided by l is constant so we can write mass this is l square divided by 12 so this is the moment of inertia for rod one so it is symmetric so moment of inertia for rod two is m l square divided by 12.

03:00 Now we know that the moment of inertia for rod third because it is center point so at center this is ml square divided by 12.

03:10 So the moment of inertia about x -axis is equal to i1 plus i2 plus i3 and all values is same so we can write three times i will so this is three times ml squared divided by 12 so this is ml squared divided by 4 so moment of inertia about x -axis is ml squared divided by 4.

03:34 So this is our moment of inertia...

Question

Please give Ace some feedback