chapter10

Angular Momentum: A vector force that is conserved in a rotating object. Since angular momentum is conserved in the absence of external torques, in the problems that invole it, the inital angular momentum is equal to the final angular momentum. - What is going to be covered (Click links if you wish to go to a certain section):  Right Hand Rule For Cross Products  How to Multiply Cross Products  Angular Momentum of a Particle  Angular Momentum of a System Rotating About a Symmetry Axis  Newton's Second Law of Angular Motion  Kinetic Energy of a Rotating Object  Torque   --  __Right Hand Rule For Cross Products __ __﻿ __The Right Hand Rule for cross products is appliued to find the direction of the resultant value (torque or angular momentum). To do this point your fingers in the direction of the radius and curl them until they are facing the direction of the force or linear momentum. Your thumb will be pointing in the direction of the torque or angular momentum.

﻿In the image above, the a is the radius, the b is the force or linear momentum, and the resultant c is the torque or angular momentum. ﻿If you wish to return to the top of the page click here <span style="color: #000000; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">-   __<span style="background-color: #ffffff; font-family: Impact,Charcoal,sans-serif;"><span style="color: #000000; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">﻿ <span style="background-color: #ffffff; font-family: Impact,Charcoal,sans-serif;">How to Multiply Cross Products  __  <span style="color: #f57f14; font-family: Impact,Charcoal,sans-serif; font-size: 90%;">An x is not simply a multiply sign. It is a cross product. Cross products are necessary because of the vector nature of torque and angular momentum. A cross product can be performed as shown here:

<span style="color: #f57f14; display: block; font-family: Impact,Charcoal,sans-serif; font-size: 140%; text-align: left;">﻿If you wish to return to the top of the page click here <span style="background-color: #ffffff; color: #000000; display: block; font-family: Impact,Charcoal,sans-serif; font-size: 120%; text-align: left;"><span style="color: #f57f14; font-family: Impact,Charcoal,sans-serif; font-size: 90%;">﻿ - __<span style="background-color: #ffffff; font-family: Impact,Charcoal,sans-serif;">Angular Momentum of a Particle __ <span style="background-color: #ffffff; color: #e1d109; display: block; font-family: Impact,Charcoal,sans-serif; font-size: 90%; text-align: left;"><span style="background-color: #ffffff; color: #e1d109; font-family: Impact,Charcoal,sans-serif; font-size: 110%;"><span style="background-color: #ffffff; color: #e1d109; font-family: Impact,Charcoal,sans-serif; font-size: 90%;">﻿ ﻿ <span style="background-color: #ffffff; color: #e1d109; font-family: Impact,Charcoal,sans-serif;">﻿The angular momentum of a particle rotating about an axis can b﻿e found by using the equation:  <span style="background-color: #ffffff; color: #000000; display: block; font-family: Impact,Charcoal,sans-serif; font-size: 120%; text-align: left;"><span style="background-color: #ffffff; font-family: Impact,Charcoal,sans-serif;"> <span style="background-color: #ffffff; color: #e1d109; font-family: Impact,Charcoal,sans-serif; font-size: 90%;">﻿ <span style="background-color: #ffffff; color: #e1d109; font-family: Impact,Charcoal,sans-serif; font-size: 80%;">﻿This equation means the angular momentum is the cross product of the radius multiplied by the linear momentum (p = m * v). <span style="background-color: #ffffff; color: #000000; display: block; font-family: Impact,Charcoal,sans-serif; font-size: 120%; text-align: left;"> ﻿﻿If you wish to return to the top of the page click here <span style="background-color: #ffffff; color: #000000; display: block; font-family: Impact,Charcoal,sans-serif; font-size: 120%; text-align: left;">- __<span style="background-color: #ffffff; font-family: Impact,Charcoal,sans-serif; font-size: 120%;">﻿ <span style="background-color: #ffffff; font-family: Impact,Charcoal,sans-serif;">﻿<span style="background-color: #ffffff; font-family: Impact,Charcoal,sans-serif;">Angular Momentum of a System Rotating About a Symmetry Axis  __ <span style="background-color: #ffffff; color: #1df71f; display: block; font-family: Impact,Charcoal,sans-serif; font-size: 90%; text-align: left;">The angular momentum of a system rotating about an axis of symmetry can be found by using the equation: <span style="background-color: #ffffff; color: #1df71f; display: block; font-family: Impact,Charcoal,sans-serif; font-size: 90%; text-align: left;"> <span style="background-color: #ffffff; color: #1df71f; font-family: Impact,Charcoal,sans-serif; font-size: 90%;">This equation means the angular momentum is the moment of inertia of the particle multiplied by the particle's angular velocity. You can see this formula applied below:

<span style="background-color: #ffffff; color: #000000; font-family: Impact,Charcoal,sans-serif; font-size: 160%;">﻿If you wish to return to the top of the page click here <span style="background-color: #ffffff; color: #000000; display: block; font-family: Impact,Charcoal,sans-serif; font-size: 90%; line-height: 0px; overflow: hidden; text-align: center;">﻿ <span style="background-color: #ffffff; color: #000000; display: block; font-family: Impact,Charcoal,sans-serif; font-size: 120%; text-align: left;">﻿- __<span style="background-color: #ffffff; font-family: Impact,Charcoal,sans-serif;">Newton's S﻿econd Law <span style="background-color: #ffffff; color: #192bf0; font-family: Impact,Charcoal,sans-serif;">﻿ <span style="background-color: #ffffff; font-family: Impact,Charcoal,sans-serif;">of Angular Motion __ <span style="background-color: #ffffff; font-family: Impact,Charcoal,sans-serif; font-size: 90%;">Newton's Second Law (rotation) states that the sum of the torques on an object is equal to the moment of inertia of that object, multiplied by the angular acceleration of the same object. This is shown by this equation:

<span style="color: #000000; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">﻿If you wish to return to the top of the page click here <span style="color: #000000; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">-   __<span style="background-color: #ffffff; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">Kinetic Energy of a Rotating __<span style="background-color: #ffffff; font-family: Impact,Charcoal,sans-serif;">__Object__   <span style="background-color: #ffffff; color: #4092d4; font-family: Impact,Charcoal,sans-serif; font-size: 90%;">The equation for Kinetic Energy, where K is kinetic energy, //I// is moment of inertia and omega is angular velocity is shown by this equation: <span style="color: #4092d4; font-family: Impact,Charcoal,sans-serif; font-size: 90%;">﻿Kinetic energy is also the integral of angular momentum. If you take the equation for angular momentum and integrate it, you come up with the equation shown above. What use would this have? If you are given a graph of the angular momentum you know you can find the kinetic energy by taking the area under the curve. On the other hand, if you have the graph of the kinetic energy, you know the lineral momentum is the slope of the curve at any given point. In order to calculate the moment of inertia in some cases you will need to apply the parallel axis theorem. To do this, follow the formula below: <span style="color: #000000; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">﻿If you wish to return to the top of the page click here <span style="color: #000000; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">  <span style="background-color: #ffffff; color: #9b06f9; display: block; font-family: Impact,Charcoal,sans-serif; text-align: left;"><span style="background-color: #ffffff; color: #9b06f9; font-family: Impact,Charcoal,sans-serif;">__Torque__  <span style="background-color: #ffffff; color: #9b06f9; display: block; font-family: Impact,Charcoal,sans-serif; text-align: left;"><span style="background-color: #ffffff; color: #9b06f9; font-family: Impact,Charcoal,sans-serif; font-size: 90%;">Torque- " <span style="background-color: #ffffff; color: #9b06f9; cursor: default; font-family: Impact,Charcoal,sans-serif;">something that produces or tends to produce torsion or <span style="background-color: #ffffff; color: #9b06f9; cursor: default; font-family: Impact,Charcoal,sans-serif;">rotation;the moment of a force or system of forces tending <span style="background-color: #ffffff; color: #9b06f9; cursor: default; font-family: Impact,Charcoal,sans-serif;">to cause <span style="background-color: #ffffff; color: #9b06f9; cursor: default; font-family: Impact,Charcoal,sans-serif;">rotation." <span style="background-color: #ffffff; color: #9b06f9; display: block; font-family: Impact,Charcoal,sans-serif; text-align: left;"><span style="background-color: #ffffff; color: #9b06f9; font-family: Impact,Charcoal,sans-serif; font-size: 90%;">The equation for Torque, where T is torque, r is radius, x is a cross product and F is the force acting on the object looks like this:

<span style="color: #000000; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">﻿If you wish to return to the top of the page click here <span style="color: #000000; font-family: Impact,Charcoal,sans-serif; font-size: 110%;">-- <span style="background-color: #ffffff; color: #000000; font-family: Impact,Charcoal,sans-serif;">[] <span style="background-color: #ffffff; color: #000000; font-family: Impact,Charcoal,sans-serif;">[]  <span style="background-color: #ffffff; color: #000000; font-family: Impact,Charcoal,sans-serif;">[]  <span style="background-color: #ffffff; color: #000000; font-family: Impact,Charcoal,sans-serif;">[]  <span style="background-color: #ffffff; color: #000000; font-family: Impact,Charcoal,sans-serif;">[]  <span style="display: block; font-family: Impact,Charcoal,sans-serif;">[]


 * CATEGORY || SCORE (1-4) || POINTS (0-20) ||
 * Content || 3 || 15 ||
 * Organization || 4 || 20 ||
 * Accuracy || 4 || 20 ||
 * Appearance || 4 || 20 ||
 * Participation || 4 || 20 ||
 * TOTAL || 95 ||