SC PHYS 201 - Rotational Work (3 pages)

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Rotational Work



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Phys 201 1nd Edition Lecture 21 Outline of Last Lecture I Torque and inertia Outline of Current Lecture II Conservation of Angular Momentum III Torque and Angular Momentum IV Rotational Work V Rotational Kinetic Energy Current Lecture Conservation of Angular Momentum There are certain instances in which the rotational velocity of an object changes but its angular momentum stays the same This is because the radius of an object is inversely correlated with the velocity of the object Remember that the angular momentum L of an object is written as L mVR This means that if we isolate the velocity of the object the equation will read V L mR This means that as the radius of the object becomes smaller the velocity will increase and vice versa The most common example of the conservation of angular momentum is a figure skater performing a spin When the skater spins with their arms extended the rotate much slower than when they pull their arms in even though no extra work is applied to make them go faster This is because the skater has a larger radius with their arms extended than with their arms pulled close to their body Torque and Angular momentum Remember that angular momentum is a vector and therefore has both direction and magnitude both of which are affected by torque In relation to angular momentum the equation for torque is Tnet L t This means that the direction of the change in angular momentum is the same as the direction of the torque Torque can increase or decrease the angular momentum of an object but it can also cause precession Precession is a change in the orientation of the axis of a rotating object Precession is best demonstrated by the behavior of These notes represent a detailed interpretation of the professor s lecture GradeBuddy is best used as a supplement to your own notes not as a substitute gyroscopes like a spinning top In this case the weight of the top is directed downwards from its center and the normal force of a surface is pushing upwards on it at the point of contact These opposite forces produce a torque perpendicular to the angular momentum which causes the top to precess The direction of the torque is changed but not its magnitude Rotational work Remember that work is the amount of force required to move an object a certain distance and that torque is the amount of force required to change the angular momentum of an object It only stands to reason that torque and work are connected The following equations demonstrate the connection between rotational work and torque W FS W Fr T Fr W T Therefore work done on a rotating object is equal to the torque force times the angle of rotation change in position Because work measures a change in energy we can use this equation to help us figure out the change in energy of a rotating object Rotational Kinetic Energy For linear movement kinetic energy KE is equal to 5 time the mass m of the object times the square of the velocity V Rotational kinetic energy is similar except we use the moment of inertia I instead of mass and the angular velocity instead of velocity KE linear 5mV2 KE rotational 5I 2 We can also use this concept to figure out the angular acceleration of a rotating object in conjunction with the kinetic energy Ke 5I 2 KE T KE I Total Kinetic energy Think when you roll a ball across a surface The ball is spinning around its axis and therefore has rotational movement However as it moves across the ground it has linear movement as well If we want to calculate the total kinetic energy of the ball we need to take both the rotational and linear kinetic energy into consideration This means that KEtotal KErotational KElinear KEtotal 5 I 2 5 mV2 This means that when we use conservation of energy with a rotational object our kinetic energy will be equal to the sum of the rotational and linear kinetic energy of the object Think about a disk with a mass m at the top of an incline with a height h If the disk starts from rest and rolls down the incline what is its speed at the bottom This is a conservation of energy problem Remember that if the object begins at rest it has no initial kinetic energy and because it is still moving at the bottom it has no final potential energy Ei Ef KEi PEi KEf PEf 0 PEi KEf 0 PEi KEf mgh 5 I 2 5 mV2


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