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Chapter 10:Rotation of a rigid object about a fixed axis Part 2 Reading assignment: Chapter 11.1-11.3 Homework : (due Wednesday, Oct. 12, 2005): Problems: Q4, 2, 5, 18, 21, 23, 24, • Rotational motion, • Angular displacement, angular velocity, angular acceleration • Rotational energy • Moment of Inertia (Rotational inertia) • Torque • For every rotational quantity, there is a linear analog. Black board example 11.3 HW 27 (a) What is the angular speed w about the polar axis of a point on Earth’s surface at a latitude of 40°N (b) What is the linear speed v of that point? (c) What are w and v for a point on the equator? Radius of earth: 6370 km Rotational energy A rotating object (collection of i points with mass mi) has a rotational ___________ energy of 1 K R I ____ 2 Where: I mi ____ i Rotational inertia Demo: Both sticks have the same weight. Why is it so much more difficult to rotate the blue stick? Black board example 11.4 2 What is the rotational inertia? 3 1 4 Four small spheres are mounted on the corners of a frame as shown. a) What is the rotational energy of the system if it is rotated about the z-axis (out of page) with an angular velocity of 5 rad/s b) What is the rotational energy if the system is rotated about the yaxis? (M = 5 kg; m = 2 kg; a = 1.5 m; b = 1 m). Rotational inertia of an object depends on: - the ________ about which the object is rotated. - the __________ of the object. - the __________ between the mass(es) and the axis of rotation. I mi ri i 2 Calculation of Rotational inertia for ____________ ________________ objects I lim ri mi 0 i 2 mi r dm r dV 2 2 Refer to Table11-2 Note that the moments of inertia are different for different ________ of rotation (even for the same object) 1 I ML2 3 I 1 ML2 12 1 I MR 2 2 Rotational inertia for some objects Page 227 Parallel axis theorem Rotational inertia for a rotation about an axis that is ____________ to an axis through the center of mass I CM I I CM _____ h Blackboard example 11.4 What is the rotational energy of a sphere (mass m = 1 kg, radius R = 1m) that is rotating about an axis 0.5 away from the center with w = 2 rad/sec? Conservation of energy (including rotational energy): Again: If there are no ___________________ forces: Energy is conserved. Rotational _____________ energy must be included in energy considerations! Ei E f U i Klinear,initial K rotational,initial U f Klinear, final K rotational, final Black board example 11.5 Connected cylinders. Two masses m1 (5 kg) and m2 (10 kg) are hanging from a pulley of mass M (3 kg) and radius R (0.1 m), as shown. There is no slip between the rope and the pulleys. (a) What will happen when the masses are released? (b) Find the velocity of the masses after they have fallen a distance of 0.5 m. (c) What is the angular velocity of the pulley at that moment? Torque F sin f r F f F cos f A force F is acting at an angle f on a lever that is rotating around a pivot point. r is the ______________ between F and the pivot point. This __________________ pair results in a torque t on the lever t r F sin f Black board example 11.6 Two mechanics are trying to open a rusty screw on a ship with a big ol’ wrench. One pulls at the end of the wrench (r = 1 m) with a force F = 500 N at an angle F1 = 80 °; the other pulls at the middle of wrench with the same force and at an angle F2 = 90 °. What is the net torque the two mechanics are applying to the screw? Torque t and angular acceleration a. Newton’s __________ law for rotation. Particle of mass m rotating in a circle with radius r. force Fr to keep particle on circular path. force Ft accelerates particle along tangent. Ft mat Torque acting on particle is ________________ to angular acceleration a: t Ia dW F ds W F s Definition of work: Work in linear motion: dW F ds W F s F s cos Component of force F along displacement s. Angle between F and s. Work in rotational motion: dW F ds Torque t and angular dW t ___ W t ___ displacement q. Work and Energy in rotational motion Remember work-kinetic energy theorem for linear motion: 1 1 2 2 W mv mv f i 2 2 External work done on an object changes its __________ energy There is an equivalent work-rotational kinetic energy theorem: 1 1 2 2 W 2 ___ w f 2 ___ wi External, rotational work done on an object changes its _______________energy Linear motion with constant linear acceleration, a. Rotational motion with constant rotational acceleration, a. v xf v xi a x t w f _________ x f xi 12 (vxi vxf )t q f ________________ 1 2 x f xi v xi t a x t 2 q f ____________________ vxf vxi 2ax ( x f xi ) w f ___________________ 2 2 2 Summary: Angular and linear quantities Linear motion 1 2 K m v Kinetic Energy: 2 Force: F ma Momentum: p mv Work: W F s Rotational motion Kinetic Energy: K R _________ Torque: t ______ Angular Momentum: Work: L __ W _____