Contents. Objectives Torque on an Object Rotational Kinetic Energy Yo yo Rolling on an Incline Physical Pendulum Angular Momentum and Torque Recap

Transcription

1 Physics 121 for Majors Class 21 Rotating Objects Last Class We learned to find angular momentum and torques of point masses and objects. We learned how to use torques and forces to solve problems with rolling objects. We learned methods of integration to find mass, center of mass, and moment of inertia. Today s Class Torque on an object Torque and angular momentum: bicycle wheel Section 1 Loosening a Bolt How would you apply a force on the wrench to loosen the bolt most easily? Why? 1

2 Loosening a Bolt What makes the torque larger? Let s apply a force at a random angle The force is greater The wrench is longer The force is perpendicular to the wrench Where is? Let s apply a force at a random angle Redraw to put the tails together to find. Redraw to put the tails together to find. sin Torque Method II Find the component of perpendicular to. Multiply by this to find the torque. sin sin 2

3 Torque Method III Torque is the whole force times the moment arm sin. Draw a line along and make a line from the center of rotation that s perpendicular to that line. Section 2 Rotational Kinetic sin sin Kinematics l yo-yo l Conservation of A yo-yo of mass and moment of inertia falls from rest a distance. What happens to its energy? It is converted to both translational and rotational kinetic energy. Translational kinetic energy of the cm: where is the speed of the cm. Rotational kinetic energy about the cm: where is the moment of inertia about the cm Conservation of A yo-yo of mass and moment of inertia (about its cm) falls from rest a distance. Assume a hub radius. What is its velocity?

4 Torque about the cm A yo-yo of mass and moment of inertia (about its cm) falls from rest a distance. Assume a hub radius. What is its velocity? and are positive, is negative, hence the minus sign. Torque about the cm A yo-yo of mass and moment of inertia (about its cm) falls from rest a distance. Assume a hub radius. What is its velocity? Torque about the Edge of the Hub A yo-yo of mass and moment of inertia (about its cm) falls from rest a distance. Assume a hub radius. What is its velocity? ball on an inclined plane d Plane A ball rolls a distance down an inclined plane of angle. It is a solid sphere of mass and radius. d Plane A ball rolls a distance down an inclined plane of angle. It is a solid sphere of mass and radius. Write an equation for energy conservation (about the cm). 4

5 d Plane A ball rolls a distance down an inclined plane of angle. It is a solid sphere of mass and radius. d Plane Why isn t there a translational kinetic energy term when you use rotation about the point of contact? Write an equation for energy conservation (about the point of contact). d Plane It s less confusing to do rotational kinetic energy about the cm! d Plane Find an equation for the torque on the sphere. Solve for. Find. Solution : sin Torque and force: sin sin sin sin sin sin sin physical pendulum 5

6 Take an object of a random shape, put a hole in it, and hang it on a nail. What is its period of oscillation? Assume we know the object s mass, cm and moment of inertia about its cm. (We can find it by integration.) Let be the object s mass be the distance from the hole to the cm be the moment of inertia about the cm We know the moment of inertia about the hole is: What is the torque about the hole? We know the moment of inertia about the hole is: The torque is sin, so sin sin What does the minus sign tell us? Angular acceleration is for + and + for --. sin In the small angle approximation What is the angular frequency? This coefficient is! 6

7 Know this equation!!! sin Section 3 Torque and Angular Momentum Bicycle You are riding your bike down the road and you turn your handlebars to the right. Bicycle You are riding your bike down the road and you lean to the right. What happens? What happens? Football Rotation A football in midair on the moon experiences no torque. Why? Its center of mass moves in a parabolic path. If you throw a nice spiral pass, the football just rotates about an axis lying along its initial path of travel. Football Precession If a pass isn t a perfect spiral, the football can precess in addition to rotate. The football precesses about the direction of its angular momentum with a constant angular velocity. 7

8 Football in Air Drag forces affect the motion of a football. See what happens during a LONG pass. Rotating a Book Try rotating a book around each of the three axes that are perpendicular to each face and that pass the center of mass. Top Rotation and Precession A top differs from a football in that gravity produces a torque on the top. The top can precess with a constant tipping angle while rotating. Top - Nutation A top can precess at a constant angle but it usually bobs and up down with the tipping angle varying. This is called nutation. An object must have a torque to nutate. Just know qualitatively what rotation, precession, and nutation mean. Section 4 Recap Big Ideas Angular momentum is l Torque is l Both depend on the origin of the coordinate system For a point mass where is the distance from the axis of rotation. Find from tables and parallel axis theorem. Integrate by the disk and shell methods to find total mass, cm, and. 8

9 Schedule Do Post-Class Quiz #21 Do Pre-Class Quiz #22 HW #20 is due tomorrow HW #21 is due Friday No quiz this week Exam 3 is Thursday! 9