Review Two points, A and B, are on a disk that rotates with a uniform speed about an axis. Point A is closer to the axis than point B. Which of the following is NOT true? 1. Point B has the greater tangential speed. 2. Point A has the lesser centripetal acceleration. 3. Points A and B have the same angular acceleration. 4. Point B has the greater angular speed. 5. Point A has the lesser tangential acceleration. The London Eye, which is a giant Ferris wheel, has a diameter of 135 m. It revolves at a constant rate, and takes 30 minutes to complete one revolution. What is the tangential velocity of a rider in a capsule that is located at the perimeter of the wheel? 1. 0.24 m/s 2. 0.47 m/s 3. 3.5 m/s 4. 7.1 m/s 5. None of the above You have a friend who lives in the southern part of the United States, and you live in the northern part. As the earth rotates, your tangential velocity is hers, and your angular velocity is hers. 1. greater than; equal to 2. equal to; greater than 3. greater than; less than 4. less than; greater than 5. less than; equal to A point P is at a distance R from the axis of rotation of a rigid body. The tangential speed, centripetal acceleration, and tangential acceleration of the point can be expressed as Linear Centripetal Tangential Speed Acceleration Acceleration 1. Rω Rω 2 Rα 2. Rω Rα Rω 2 3. Rω 2 Rα Rω 4. Rω Rω 2 Rω 5. Rω 2 Rα Rω 2 You give an orbiting satellite a command to rotate through an angle given by θ = at + bt 2 ct 4 where a, b, and c are constants and θ is in radians if t is in seconds. What is the angular acceleration of this satellite at time t? 1. at 2. a + b c 3. 12 4. 2b 12ct 2 5. zero Moment of Inertia Race Objects with more mass further out have larger moments of inertia Things with larger moments of inertia are more difficult to accelerate angularly Sphere, Cylinder and Ring of same mass. Rank from largest to smallest moment of Inertia. 1
Moment of Inertia Race Objects with more mass further out have larger moments of inertia Things with larger moments of inertia are more difficult to accelerate angularly The moment of inertia of a set of dumbbells, considered as two mass points m separated by a distance 2L about the axis AA, is 1. ml 2 2. ½mL 2 3. 2mL 2 4. ¼ ml 2 5. 4mL 2 A homogeneous solid cylinder of mass m, length L, and radius R rotates about an axis through point P, which is parallel to the cylinder axis. If the moment of inertia about the cylinder axis is ½mR 2, the moment of inertia about the axis through P is 1. 0.4mR 2 2. ½ mr 2 3. 2/3 mr 2 4. mr 2 5. 1.5mR 2 To increase the moment of inertia of a body about an axis, you must 1. increase the angular acceleration. 2. increase the angular velocity. 3. decrease the angular velocity. 4. make the body occupy less space. 5. place part of the body farther from the axis. Torque Torque is similar to force, but for rotating objects Torque is a force at a distance (describes how much a rigid object will rotate based on the forces applied at distances from the axis of rotation) Mathematically, torque is the product of the tangential force component times the distance from the rotation axis Important concept for medical, physical therapy and exercise science folks! Example: catching a ball puts torque on your elbow 2
What torque is applied on an object that is being pulled from its axis of rotation? What torque is applied to a wheel that has a rope wrapped around it when the rope is pulled? Newton s second law for rotation: The sum of the torques (one for each force applied at a distance) = moment of inertia times angular acceleration (remember, ΣF = ma?) The diagrams show the application of a force to a level arm. Which diagram has the largest torque about the hinge (hinge is the blue dot)? 1. a 2. b 3. c L F 2F a b c 4. They are the same. 5. Unable to determine. 2L 0.5F 0.5L Example 9-9 As you begin to pedal in cycling class (suspended wheel), the chain applies a force to the rear sprocket wheel at a distance of 7.0cm from the rotation axis of the wheel. Consider the wheel to be a hoop of radius 35cm and mass 2.4kg. What is the angular velocity of the wheel after 5.0 seconds? Example 9-10 Prep Assignment, Review Moment of Inertia Integrals A uniform thin rod of mass M and length L is pivoted at one end. It is held horizontal and released. Effects due to friction and air resistance are negligible. Find the angular acceleration of the rod just after release. 3
Begin chp 10 A new mathematical tool: the Cross Product On formula sheet, written explicitly Example you try one Vector nature of rotation Two examples of how Right Hand Rule (rhr) is used: 1. Angular velocity 2. Torque Direction: right hand rule Magnitude As a particle with a velocity in the negative x direction passes through the point (0, 0, 1), it has an angular velocity relative to the origin that is best represented by vector A torque is applied to a bolt by hanging a weight w from the end of the wrench, as shown. The coordinate axis along which the torque vector is directed is A. y B. x C. y D. x E. z A. B. C. D. E. The vector C could represent A B B A A B B A None of these is possible. A 7-kg mass and a 4-kg mass are mounted on a spindle that is free to turn about the x axis as shown. Assume the mass of the arms and the spindle to be negligible. If the system is free to rotate and is released from rest, there will initially be a resultant torque in which of the following directions? A. z B. z C. y D. x E. x 4
A wheel is rotating clockwise as shown. Its rotation axis coincides with the x-axis. A torque that causes the wheel to slow down is best represented by the vector Torque and Angular momentum Mathematical definition of angular momentum (a vector!) Angular momentum of a system about an axis How was the solar system formed? A particle of mass m is moving with a velocity v, in the yz plane as shown. The vector that most nearly represents the angular momentum about the x axis is A wheel is set spinning and is then hung by a rope placed at one end of the axle. If the wheel is spinning as shown, the angular momentum of the wheel could be represented by vector Example 10-2 Find the angular momentum about the origin for the following situations: a) A car of mass 1200 kg moves in a circle of radius 20 m (centered at the origin in the xy plane) with a speed of 15m/s (counterclockwise) b) The same car moves in the xy plane with velocity -15m/s î along the line y = y 0 = 20 m. c) A uniform disk in the xy plane of radius 20 m and mass 1200 kg rotates at 0.75 rad/s about its axis, which is also the z axis. 5
Newton s second law, again! Net torque is related to angular momentum just as the net force was related to momentum Example 10-3 An Atwood s machine has two blocks with masses m1 and m2 (m1 > m2) connected by a string of negligible mass that passes over a pulley with frictionless bearings. The pulley is a uniform disk of mass M and radius R. The string does not slip on the pulley. Find the angular acceleration of the pulley and the linear acceleration of the blocks. Conservation of Angular momentum If the net external torque acting on a system about some point is zero, the angular momentum of the system about that point remains constant Example 10-4 Disk 1 is rotating freely and has an angular velocity ω i about a vertical axis that coincides with its symmetry axis. Its moment of inertia about this axis is I 1. It drops onto disk 2, of moment of inertia I 2, that is initially at rest. Disk 2 is centered on the same axis as disk 1. Because of kinetic friction, the two disks eventually attain a common angular velocity ω f. Find ω f 6