Overview
6.01A - represent rotational motion using graphs, written descriptions, and equations; apply mathematical models for constant angular acceleration to solve for unknown values; translate between rotational and translational motion
6.02B - use the rotational version of Newton's 2nd law to connect torque, angular acceleration, and moment of inertia for accelerating or static situations; use center of mass to determine how the object will translate or rotate.
6.03A - draw an extended free body diagram.
6.04B - calculate or compare torques given a force, lever arm, and angle; determine direction of torque about any point, including gravitational torque.
6.05A - describe qualitatively how changing the mass or distribution of mass affects the rotational inertia; understand that rotational inertia is a property of an object
6.06B - recognize the impact of mass, its distribution and angular velocity in the magnitude of rotational kinetic energy; include this when applying concepts of conservation of energy
6.07B - recognize that a change in momentum is caused by an unbalanced external torque applied over a time interval
6.08B - apply conservation of angular momentum qualitatively or quantitatively to determine changes in angular velocity or rotational inertia, torques or other relevant information
6.09B - solve advanced problems that involve calculations of rotational energy, angular momentum and concepts of energy, forces, momentum, and motion.
6.01A - represent rotational motion using graphs, written descriptions, and equations; apply mathematical models for constant angular acceleration to solve for unknown values; translate between rotational and translational motion
6.02B - use the rotational version of Newton's 2nd law to connect torque, angular acceleration, and moment of inertia for accelerating or static situations; use center of mass to determine how the object will translate or rotate.
6.03A - draw an extended free body diagram.
6.04B - calculate or compare torques given a force, lever arm, and angle; determine direction of torque about any point, including gravitational torque.
6.05A - describe qualitatively how changing the mass or distribution of mass affects the rotational inertia; understand that rotational inertia is a property of an object
6.06B - recognize the impact of mass, its distribution and angular velocity in the magnitude of rotational kinetic energy; include this when applying concepts of conservation of energy
6.07B - recognize that a change in momentum is caused by an unbalanced external torque applied over a time interval
6.08B - apply conservation of angular momentum qualitatively or quantitatively to determine changes in angular velocity or rotational inertia, torques or other relevant information
6.09B - solve advanced problems that involve calculations of rotational energy, angular momentum and concepts of energy, forces, momentum, and motion.
Center of Mass
For everything we've learned this year, we have treated everything as a single point at the center of mass. We have ignored extended mass beyond the object's center. This works well for a lot of predictions, but it is an over simplification. Center of Mass - a point representing the mean position of the matter in a body or system. Point Object - assume all mass is located at a single point Extended Object - assume mass is distributed throughout the object How to find the center of mass? There are two ways to find the center of mass of an object. Hang Test - hang the object from a point, draw a vertical line. Hang the object again and then draw another vertical line. Where they intersect is the center of mass. (See Figure 1) Equation - we can also find the center of mass by using an equation (See Figure 2) |
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Rotational Kinematics
- The following video explains the way we can determine the direction of the rotation - RIGHT HAND RULE
Variables
Torque
Torque and the Position of Force
Torque Equation |
Static Equilibrium
- Net Torque = 0
- Net Force = 0
Rotational Inertia & Rotational Energy
Rotational Inertia (I)
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*Note that when discussing the change in rotational Inertia, radius is much more important than mass because according to the equation:
- I = mr^2
How does rotational inertia relate to Newton's 2nd Law?
Rotational Kinetic Energy
- Newton’s 2nd law relates force to acceleration. In the angular version of Newton’s 2nd law, torque τ takes the place of force and rotational inertia(I) takes the place of mass. When the rotational inertia of an object is constant, the angular acceleration is proportional to torque.
- Net F = m*a
- Net τ =I*α
Rotational Kinetic Energy
- Rotational Kinetic Energy of an object with rotational inertia turning with rotational speed is:
- K Rotational = 1/2I*w^2
- (K Linear = 1/2m*v^2)
Angular Momentum
Linear Momentum V.S. Angular Momentum
Linear Momentum V.S. Angular Momentum
Conservation of Angular Momentum
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Spinning Chair Example proves the conservation of rotational momentum
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