Rotation and Torque Lecture 09 Thursday 12 February
![Rotation and Torque Lecture 09 Thursday: 12 February 2004 Rotation and Torque Lecture 09 Thursday: 12 February 2004](https://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-1.jpg)
Rotation and Torque Lecture 09 Thursday: 12 February 2004
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![ROTATION: DEFINITIONS • Angular position: • Angular displacement: q q 2 – q 1 ROTATION: DEFINITIONS • Angular position: • Angular displacement: q q 2 – q 1](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-3.jpg)
ROTATION: DEFINITIONS • Angular position: • Angular displacement: q q 2 – q 1 = Dq w w Instantaneous Angular velocity:
![What is the direction of the angular velocity? • Use your right hand • What is the direction of the angular velocity? • Use your right hand •](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-4.jpg)
What is the direction of the angular velocity? • Use your right hand • Curl your fingers in the direction of the rotation • Out-stretched thumb points in the direction of the angular velocity
![DEFINITIONS (CONTINUED) DEFINITIONS (CONTINUED)](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-5.jpg)
DEFINITIONS (CONTINUED)
![Direction of Angular Acceleration The easiest way to get the direction of the angular Direction of Angular Acceleration The easiest way to get the direction of the angular](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-6.jpg)
Direction of Angular Acceleration The easiest way to get the direction of the angular acceleration is to determine the direction of the angular velocity and then… • If the object is speeding up, velocity and acceleration must be in the same direction. • If the object is slowing down, velocity and acceleration must be in opposite directions.
![For constant a wx®q wv® wa®a For constant a wx®q wv® wa®a](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-7.jpg)
For constant a wx®q wv® wa®a
![Relating Linear and Angular Variables Relating Linear and Angular Variables](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-8.jpg)
Relating Linear and Angular Variables
![Three Accelerations 1. Centripetal Acceleration (radial component of the linear acceleration) -always non-zero in Three Accelerations 1. Centripetal Acceleration (radial component of the linear acceleration) -always non-zero in](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-9.jpg)
Three Accelerations 1. Centripetal Acceleration (radial component of the linear acceleration) -always non-zero in circular motion. 2. Tangential Acceleration (component of linear acc. along the direction of the velocity) -non-zero if the object is speeding up or slowing down. 3. Angular Acceleration (rate of change in angular velocity) -non-zero is the object is speeding up or slowing down.
![Energy Considerations Although its linear velocity v is zero, the rapidly rotating blade of Energy Considerations Although its linear velocity v is zero, the rapidly rotating blade of](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-10.jpg)
Energy Considerations Although its linear velocity v is zero, the rapidly rotating blade of a table saw certainly has kinetic energy due to that rotation. How can we express the energy? We need to treat the table saw (and any other rotating rigid body) as a collection of particles with different linear speeds.
![KINETIC ENERGY OF ROTATION KINETIC ENERGY OF ROTATION](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-11.jpg)
KINETIC ENERGY OF ROTATION
![Defining Rotational Inertia • The larger the mass, the smaller the acceleration produced by Defining Rotational Inertia • The larger the mass, the smaller the acceleration produced by](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-12.jpg)
Defining Rotational Inertia • The larger the mass, the smaller the acceleration produced by a given force. • The rotational inertia I plays the equivalent role in rotational motion as mass m in translational motion. • I is a measure of how hard it is to get an object rotating. The larger I, the smaller the angular acceleration produced by a given force.
![Determining the Rotational Inertia of an Object I is a function of both the Determining the Rotational Inertia of an Object I is a function of both the](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-13.jpg)
Determining the Rotational Inertia of an Object I is a function of both the mass and shape of the object. It also depends on the axis of rotation. 1. For common shapes, rotational inertias are listed in tables. A simple version of which is in chapter 11 of your text book. 2. For collections of point masses, we can use : where r is the distance from the axis (or point) of rotation. 3. For more complicated objects made up of objects from #1 or #2 above, we can use the fact that rotational inertia is a scalar and so just adds as mass would.
![](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-14.jpg)
![Comparison to Translation • x®q • v® • a®a • m®I • K=1/2 mv Comparison to Translation • x®q • v® • a®a • m®I • K=1/2 mv](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-15.jpg)
Comparison to Translation • x®q • v® • a®a • m®I • K=1/2 mv 2 1/2 I 2
![](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-16.jpg)
![Force and Torque Force and Torque](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-17.jpg)
Force and Torque
![Torque as a Cross Product (Like F=Ma) The direction of the Torque is always Torque as a Cross Product (Like F=Ma) The direction of the Torque is always](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-18.jpg)
Torque as a Cross Product (Like F=Ma) The direction of the Torque is always in the direction of the angular acceleration. • For objects in equilibrium, =0 AND F=0
![Torque Corresponds to Force • Just as Force produces translational acceleration (causes linear motion Torque Corresponds to Force • Just as Force produces translational acceleration (causes linear motion](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-19.jpg)
Torque Corresponds to Force • Just as Force produces translational acceleration (causes linear motion in an object starting at rest, for example) • Torque produces rotational acceleration (cause a rotational motion in an object starting from rest, for example) • The “cross” or “vector” product is another way to multiply vectors. Cross product results in a vector (e. g. Torque). Dot product (goes with cos ) results in a scalar (e. g. Work)
![An Example x W Forces on “extended” bodies can be viewed as acting on An Example x W Forces on “extended” bodies can be viewed as acting on](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-20.jpg)
An Example x W Forces on “extended” bodies can be viewed as acting on a point mass (with the same total mass) At the object’s center of mass (balancing point)
![Determining Direction of A CROSS PRODUCT Determining Direction of A CROSS PRODUCT](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-21.jpg)
Determining Direction of A CROSS PRODUCT
![Angular Momentum of a Particle • Angular momentum of a particle about a point Angular Momentum of a Particle • Angular momentum of a particle about a point](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-22.jpg)
Angular Momentum of a Particle • Angular momentum of a particle about a point of rotation: • This is similar to Torques
![Find the direction of the angular momentum vector-Right hand rule P r r P Find the direction of the angular momentum vector-Right hand rule P r r P](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-23.jpg)
Find the direction of the angular momentum vector-Right hand rule P r r P
![Does an object have to be moving in a circle to have angular momentum? Does an object have to be moving in a circle to have angular momentum?](http://slidetodoc.com/presentation_image/3adbb5e2a64f67fc6e21f9fbc2a2575d/image-24.jpg)
Does an object have to be moving in a circle to have angular momentum? • No. • Once we define a point (or axis) of rotation (that is, a center), any object with a linear momentum that does not move directly through that point has an angular momentum defined relative to the chosen center as
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- Slides: 26