Rotational Kinematics and Energy (Cont.) Chapter 10 Rotational Kinematics and Energy (Cont.) Dr. Jie Zou PHY 1151G Department of Physics
Dr. Jie Zou PHY 1151G Department of Physics Outline Connections between linear and rotational quantities (1) Tangential speed (2) Centripetal acceleration and tangential acceleration Real world examples (BIO) Rolling motion Rotational kinetic energy and the moment of inertia Conservation of energy Dr. Jie Zou PHY 1151G Department of Physics
Dr. Jie Zou PHY 1151G Department of Physics (1) Tangential speed Tangential speed v: The speed of something moving along a circular path; the direction of motion is always tangential to the circle; SI units: m/s. Relation between tangential and angular speed: v = r . : The angular speed; it must be in rad/s. is the same for every point on a rotating object; v is greater on the outside of a rotating object than inside and closer to the axis. Dr. Jie Zou PHY 1151G Department of Physics
(2) Centripetal and tangential acceleration Centripetal acceleration acp: The acceleration due to changing direction of motion. Magnitude: acp = v2/r = r2; SI unit: m/s2 Direction: Directed towards the axis of rotation Tangential acceleration at: The acceleration due to changing angular speed. Magnitude: at = r; SI unit: m/s2 Direction: Tangential to the circular path Microhematocrit centrifuge Dr. Jie Zou PHY 1151G Department of Physics
Real world example (BIO) Example 10-3 The microhematocrit: In a microhematocrit centrifuge, small samples of blood are placed in heparinized capillary tubes (heparin is an anticoagulant). The tubes are rotated at 11,500 rpm, with the bottom of the tubes 9.07 cm from the axis of rotation. (a) Find the linear speed of the bottom of the tubes. (b) What is the centripetal acceleration at the bottom of the tubes? Dr. Jie Zou PHY 1151G Department of Physics
Rolling motion The linear (or translational) speed of the axle of a rolling object: v = 2r /T = r = vt A rolling object combines rotational motion and translation motion: (a) pure rotational motion; (b) pure translational motion; (c) rolling without slipping (a) (b) (c)
Dr. Jie Zou PHY 1151G Department of Physics Example 10-6 A car with tires of radius 32 cm drives on the highway at 55 mph. (a) What is the angular speed of the tires? (b) What is the linear speed of the top of the tires (1 mph = 0.447 m/s). Answers: (a) 77 rad/s. (b) 110 mph. Dr. Jie Zou PHY 1151G Department of Physics
Rotational kinetic energy and the moment of inertia SI unit: J I: Moment of inertia (SI unit: kg·m2) : Angular speed (SI unit: rad/s) Moment of inertia, I I = miri2 SI unit: kg·m2 Moment of inertia, I, depends on: Distribution of mass Location and orientation of the axis of rotation The moment of inertia plays the same role in rotational motion that mass plays in translational motion. Dr. Jie Zou PHY 1151G Department of Physics
Examples: moment of inertia (a) Use the general definition, find the moment of inertia, I, for the dumbbell-shaped object. (b) If the two masses are moved closer to the axis of rotation, such that each has a radius of R/2, what is I now? (c) If the object is rotated about one end, what is its moment of inertia, I, about this new axis of rotation? (a) (c) Dr. Jie Zou PHY 1151G Department of Physics
Table 10-1 Moments of inertia for objects of various shapes Dr. Jie Zou PHY 1151G Department of Physics
Dr. Jie Zou PHY 1151G Department of Physics Real world examples Dr. Jie Zou PHY 1151G Department of Physics
Application of conservation of mechanical energy to a rolling object Kinetic energy of rolling motion: The kinetic energy of a rolling object is the sum of its translational kinetic energy and its rotational kinetic energy. Example: An object of mass m, radius r, and moment of inertia I at the top of a ramp is released from rest and rolls without slipping to the bottom, a vertical height h below the starting point. What is the object’s speed on reaching the bottom?
Dr. Jie Zou PHY 1151G Department of Physics Homework See online homework assignment on www.masteringphysics.com Dr. Jie Zou PHY 1151G Department of Physics