1Chapter 8Rotational Motion22Let’s Put a Spin on Things• Every thing about motion we have learned so far has been about objects moving in straight paths– This is called linear motion• Now we want to talk about how objects move in a circular path– This is called circular or angular motion33Rotational Speed• Before, speed was defined to be the distance traveled in a unit of time• Rotational speed (ω) is defined to be the number of rotations or revolutions an object goes through in a unit of time • The closer a point to the edge of a rotating object the greater its tangential speedRotational speed is measured in units of radians per second.There are 2π radians in 360oor one rotation.44Tangential VelocityThe tangential velocity is always at a right angle to the radius of the rotation.The closer you are to the center of the rotation, the slower your tangential velocity.55Rotational Inertia• Just like Newton’s 1st Law stated that mass is a measure of inertia, there is a rotational analog• “An object rotating about an axis tends to remain rotating about the same axis unless interfered with by some external influence”• The farther away the mass is concentrated from the center of rotation, the greater the rotational inertia– Rotation inertia is also called moment of inertia (I)Rotational inertia is the rotational analog to mass in linear motion.66Problem Solving: Rotational Inertia• Which object makes it to the bottom of a ramp first, a solid disk or a hoop?–The disk has more of its mass near its rotational axis and therefore has less rotational inertia than the hoop. Therefore, the disk will beat the hoop down the inclined plane.This was demonstrated in class.77Torque• Torque is the rotational analog to force:–Units: Nm• The lever arm is the shortest distance between where the force is being applied and the rotational axisforce armlever torque ×=Emphasize that the lever arm is very important in rotational motion.Using a pipe to extend the pipe’s length makes it easier to loosen a bolt.88Problem Solving: Torque• You need to apply a fixed amount of torque to a door to open it. How much more force will you need to apply when you have a lever arm of 0.1 m compared to when your lever arm is 1 m?–Remember: torque = F*lever arm, so if we reduce the lever arm by 10, we must increase the force by 10. Therefore we need 10 times the force to open the door.99Center of Mass•The center of mass (CM) is the average mass point– You can think of it as the point that divides the mass in two• An object always rotates about its CM• The CM follows a straight line path (or whatever path a non-rotating object would follow)– In problems, the center of mass is used as the object’s point location1010Centripetal Force• Any force directed toward a fixed center is called a centripetal forceNote that the acceleration is always directed toward the center of rotation.Since this force is perpendicular to the tangential velocity, it does not slow down or speed up the object. This force only serves to change the direction on the object.1111Centrifugal “Force”• Centrifugal “force” is attributed an object being pushed away from the center• But the centrifugal “force” is NOT a force at all– If the centripetal force is removed, the object continues on in a straight pathWe demonstrated that water in a bucket will stay in a bucket when swung overhead (given that you swing it fast enough).1212Rotating Reference Frames• Centrifugal “forces” can be used to simulate the feeling of gravity• Consider to futuristic space colony shaped like an inner tube• Given the radius of the inner tube, a ω can be chosen to make the centrifugal “force” feel like gravityYou can see a conception of this space station in the move “2010.”1313Problem Solving: Rotating Frames• If the Earth suddenly stopped rotating, how would your weight change?– A. It will be greater– B. It will be less– C. There will be no changeA. Since rotating objects “want” to travel in a straight line, we would be flung form the Earth were it not for gravity. If the Earth stops rotating, we will no longer “want” to move away from the Earth, gravity will feel stronger and our weight will increase.1414Angular Momentum• Angular momentum is the analog to linear momentum:•Newton’s 2ndLaw states that objects in motion tend to stay in motion• The same applies in rotational motion– “An object or system of objects will maintain its angular momentum unless acted upon by an unbalanced external torque”ωI momentumangular =Note that linear momentum = mv (here m -> I, v -> ω)1515Conservation of Angular Momentum• Angular momentum is conserved just like linear momentum• A decrease in I will increase the ω• Linear momentum can be converted into angular momentum– The total linear momentum and angular momentum is always conserved!We did the experiment where someone sat on a chair that could easily rotate and, once the person was sent rotating, they would move their arms out holding weights and they would slow down.1616Problem Solving: Angular Momentum• Someone is sitting on a stool that can freely rotate but is currently at rest. What will happen if you throw a bean bag at them and they catch it? –The momentum of the bean bag before it is caught will be transferred into angular momentum, and the person hold the bag will begin to rotate.This was then demonstrated at the end of