PHYSICS 231 INTRODUCTORY PHYSICS I Lecture 12 Last Lecture GMm grav R2 G 6 67 10 11 Nm 2 kg 2 Newton s Law of gravitation F Kepler s Laws of Planetary motion 1 Ellipses with sun at focus 2 Sweep out equal areas in equal times 3 R GM 3 Constant 2 2 T 4 Gravitational Potential Energy PE mgh valid only near Earth s surface For arbitrary altitude Mm PE G r Zero reference level is at r Example 7 18 You wish to hurl a projectile from the surface of the Earth Re 6 38x106 m to an altitude of 20x106 m above the surface of the Earth Ignore rotation of the Earth and air resistance a 9 736 m s a What initial velocity is required b What velocity would be required in order for the projectile to reach infinitely high b 11 181 m s I e what is the escape velocity c skip How does the escape velocity c 7 906 m s compare to the velocity required for a low earth orbit 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 meters Door Demo 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 r r F F Non perpendicular forces Fr sin is the angle between F and r Torque and Equilibrium Forces sum to zero no linear motion Fx 0 and Fy 0 Torques sum to zero no rotation 0 Axis of Rotation Torques require point of reference Point can be anywhere Use same point for all torques Pick the point to make problem easiest eliminate unwanted Forces from equation 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 one can treat gravity as if it acts at one point the center ofgravity m x i i M tot gX mi g x i M tot g M tot mx X i Center of gravity M tot 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 hat 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 hat point should I use for torque origin A B C D Example 8 4c Tleft Wbeam B A Given Tleft 250 Tright 400 Find Wbox Tright D C 8m 2m Wbox hat point should I use for torque origin A B C D Example 8 4d Tleft Wbeam B A Given Wbeam 300 Wbox 200 Find Tright Tright D C 8m 2m Wbox hat 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 hat point should I use for torque origin A B C D Torque and Angular Acceleration Analogous to relation between F and a F ma I Moment of Inertia F m R FR mat R m R R mR 2 Moment of Inertia Moment of inertia I rotational analog to mass I mi ri2 i r defined relative to rotation axis SI units are kg m2 Baton Demo Moment of Inertia Demo More About Moment of Inertia Depends on 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 M c What is the acceleration of the bucket 2 0 9 kg m2is b c 4 m s2 d What the6 67 massrad s of the bucket d 1 72 kg Example 8 9 A 600 kg solid cylinder of radius which can rotate freely about its accelerated by hanging a 240 kg mass from the string which is wrapped about the 0 6 m axis is end by a cylinder a Find the linear acceleration of4 36 the m s mass 2 b What is the speed of the mass after it has dropped 2 5 m 4 67 m s
View Full Document
Unlocking...