Chapter 8 Rotational Equilibrium and Rotational Dynamics Wrench Demo 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 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 Non perpendicular forces Fr sin is the angle between F and r Torque and Equilibrium Fx 0 and Fy 0 Forces sum to zero no linear motion 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 Pick the point to make problem least difficult Example 8 1 Given M 120 kg Neglect the mass of the beam a Find the tension in the cable b What is the force between the beam and the wall a T 824 N b f 353 N Another Example Given W 50 N L 0 35 m x 0 03 m Find the tension in the muscle W x L F 583 N Center of Gravity 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 ofgravity X mi g xi mi i i Weighted Average mi xi i mi g i mi xi MgX whereX i mi i Example 8 2 Given x 1 5 m L 5 0 m wbeam 300 N wman 600 N Find T Fig 8 12 p 228 Slide 17 T 413 N x L Example 8 3 Consider the 400 kg beam shown below Find TR TR 1 121 N Example 8 4a Tleft Wbeam B A Given Wbeam 300 Wbox 200 Find Tleft Tright D C 8m 2m Wbox What point should I use for torque origin A B C D Example 8 4b Tleft Wbeam B A Given Tleft 300 Tright 500 Find Wbeam Tright D C 8m 2m Wbox What point should I use for torque origin A B C D Example 8 4c Tleft Wbeam B A Given Wbeam 300 Wbox 200 Find Tright Tright D C 8m 2m Wbox What point should I use for torque origin A B C D Example 8 4d Tleft Wbeam B A Given Tleft 250 Tright 400 Find Wbox Tright D C 8m 2m Wbox What point should I use for torque origin A B C D Example 8 4e Tleft Wbeam B A Given Tleft 250 Wbeam 250 Find Wbox Tright D C 8m 2m Wbox What point should I use for torque origin A B C D Example 8 5 skip 80 kg beam of length L 100 cm has a 40 kg mass anging from one end At what position x can one balan hem beam at a point L 100 cm 80 kg x x 66 67 cm 40 kg Baton Demo Moment of Inertia Demo Torque and Angular Acceleration Analogous to relation between F and a F ma I Moment of Inertia Moment of Inertia Mass analog is moment of inertia I I mi ri2 i r defined relative to rotation axis SI units are kg m2 More About Moment of Inertia I depends on both the mass and its distribution If mass is distributed further from axis of rotation moment of inertia will be larger Moment of Inertia of a Uniform Ring Divide ring into segments The radius of each segment is R I mi ri2 MR2 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 about the z axis a 0 72 kg m2 b 1 08 kg m2 c 1 8 kg m2 Other Moments of Inertia Other Moments of Inertia cylindrical shell I MR2 bicycle rim 1 solid cylinder I MR2 filled can of coke 2 1 baton rod about center I ML2 12 1 2 baseball bat rod about end I ML 3 2 2 basketball spherical shell I MR 3 2 boulder solidsphere I MR2 5 Example 8 7 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 8 skip 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 b a 10 8 m s2 Example 8 9 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 dropped 2 5 m 4 67 m s Rotational Kinetic Energy Each point of a rigid body rotates with angular velocity 1 1 2 2 2 KE mi vi mi ri 2 2 1 2 KE I 2 Including the linear motion 1 2 1 2 KE mv I 2 2 KE due to rotation KE of center of mass motion Example 8 10 What is the kinetic energy of the Earth due to the daily rotation Given Mearth 5 98 x1024 kg Rearth 6 36 x106 m 2 56 x1029 J 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 v 23 7 m s Demo Moment of Inertia Olympics Example 8 12a The winner is A Hollow Cylinder B Solid Cylinder Example 8 12b The winner is A Hollow Cylinder B Sphere Example 8 12c The winner is A Sphere B Solid Cylinder Example 8 12d The winner is A Solid Cylinder B Mountain Dew Example 8 12e The winner is A Sphere B Mountain Dew Angular Momentum Rigid body L I L mvr m r 2 Point particle Analogy between L and p Angular Momentum Linear momentum L I p mv L t F p t Conserved if no net Conserved if no net outside torques outside forces Rotating Chair Demo 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 A r v t 2 …
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