Wrench Demo Chapter 8 Rotational Equilibrium and Rotational Dynamics Torque Torque is tendency of a force to rotate object about some axis Fd F is the force d is the lever arm or moment arm Units are Newton m Non perpendicular forces Torque is vector quantity Direction determined by axis of twist Perpendicular to both r and F Clockwise torques point into paper Defined as negative Counter clockwise torques point out of paper Defined as positive Torque and Equilibrium Fx 0 and Fy 0 Fr sin Forces sum to zero no linear motion is the angle between F and r Torques sum to zero no rotation 0 Meter Stick Demo Axis of Rotation Torques require point of reference Point can be anywhere Use same point for all torques Example 8 1 Pick the point to make problem least difficult Another Example Given M 120 kg Neglect the mass of the beam Given W 50 N L 0 35 m x 0 03 m Find the tension in the muscle W x a Find the tension in the cable L F 583 N b What is the force between the beam and the wall a T 824 N b f 353 N Center of Gravity Example 8 2 Gravitational force acts on all points of an extended object However it can be considered as one net force acting on one point the center of gravity X mi g xi mi i i Weighted Average mi xi i mi g Given x 1 5 m L 5 0 m wbeam 300 N Find T wman 600 N T 413 N i MgX where X mi xi i mi i x L Example 8 4a Example 8 3 Consider the 400 kg beam shown below Find TR Given Wbeam 300 Wbox 200 Find Tleft What point should I use for torque origin TR 1 121 N Example 8 4b Example 8 4c Given Tleft 300 Given Wbeam 300 Tright 500 Find Wbeam Wbox 200 Find Tright What point should I use for torque origin A B C D Example 8 4d A B C D Given Tleft 250 Tright 400 What point should I use for torque origin What point should I use for torque origin Example 8 4e Given Tleft 250 Find Wbox A B C D Wbeam 250 A B C D Find Wbox What point should I use for torque origin A B C D Example 8 5 skip A 80 kg beam of length L 100 cm has a 40 kg mass hanging from one end At what position x can one balance them beam at a point L 100 cm Baton Demo Moment of Inertia Demo 80 kg x 40 kg x 66 67 cm Torque and Angular Acceleration Moment of Inertia Analogous to relation between F and a F ma I mi ri2 I i Moment of Inertia More About Moment of Inertia Mass analog is moment of inertia I r defined relative to rotation axis SI units are kg m2 Moment of Inertia of a Uniform Ring I depends on both the mass and its distribution If mass is distributed further from axis of rotation moment of inertia will be larger Divide ring into segments The radius of each segment is R I mi ri2 MR 2 Other Moments of Inertia Example 8 6 What is the moment of inertia of the following point masses arranged in a square a about the x axis b about the y axis c a b c about the z axis 0 72 kg m2 1 08 kg m2 1 8 kg m2 Other Moments of Inertia cylindrical shell I MR 1 solid cylinder I MR 2 2 1 rod about center I ML2 12 1 rod about end I ML2 3 2 spherical shell I MR 2 3 2 solidsphere I MR 2 5 2 bicycle rim filled can of coke baton baseball bat basketball boulder Example 8 8 skip Treat the spindle as a solid cylinder a What is the moment of Inertia of the spindle M 5 0 kg R 0 6 m b If the tension in the rope is 10 N what is the angular acceleration of the wheel c What is the acceleration of the bucket M d What is the mass of the bucket a 0 9 kg m2 b 6 67 rad s2 c 4 m s2 d 1 72 kg Example 8 9 A cylindrical space station of R 12 M 3400 kg has moment of inertia 0 75 MR2 Retrorockets are fired tangentially at the surface of space station and provide impulse of 2 9x104 N s a What is the angular velocity of the space station after the rockets have finished firing b What is the centripetal acceleration at the edge of the space station a 0 948 rad s Example 8 7 b a 10 8 m s2 A 600 kg solid cylinder of radius 0 6 m which can rotate freely about its axis is accelerated by hanging a 240 kg mass from the end by a string which is wrapped about the cylinder a Find the linear acceleration of the mass 4 36 m s2 b What is the speed of the mass after it has 4 67 m s dropped 2 5 m Example 8 10 Rotational Kinetic Energy Each point of a rigid body rotates with angular velocity 1 1 mi vi2 mi ri2 2 2 2 1 KE I 2 2 KE What is the kinetic energy of the Earth due to the daily rotation Given Mearth 5 98 x1024 kg Rearth 6 36 x106 m Including the linear motion KE 1 2 1 2 mv I 2 2 2 56 x1029 J KE due to rotation KE of center of mass motion Example 8 11 A solid sphere rolls down a hill of height 40 m What is the velocity of the ball when it reaches the bottom Note We don t know R or M Demo Moment of Inertia Olympics v 23 7 m s Example 8 12a Example 8 12b The winner is The winner is A Hollow Cylinder A Hollow Cylinder B Solid Cylinder B Sphere Example 8 12c Example 8 12d The winner is The winner is A Sphere A Solid Cylinder B Solid Cylinder B Monster Example 8 12e Angular Momentum The winner is A Sphere L mvr m r 2 B Mountain Dew Rotating Chair Demo Rigid body L I Point particle Analogy between L and p Angular Momentum Linear momentum L I p mv L t F p t Conserved if no net outside torques Conserved if no net outside forces Angular Momentum and Kepler s 2nd Law For central forces e g gravity 0 and L is conserved Change in area in t is 1 r v t 2 L mrv A A 1 L t 2m Example 8 14 Example 8 13 Two twin ice skaters separated by 10 meters skate without friction in a circle by holding onto opposite ends of a rope They move around a circle once every five seconds By reeling in the rope they approach each other until …
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