3 Dimensional Rotational Motion and Gyroscopes 8 01 W14D2 Today s Reading Assignment MIT 8 01 Course Notes Chapter 22 Three Dimensional Rotations and Gyroscopes Sections 22 4 Announcements Problem Set 12 Due Week 14 Thursday 9 pm Math Review Week 15 Tuesday at 9 pm in 26 152 Sunday Tutoring in 26 152 from 1 5 pm Final Exam Dec 16 Johnson Athletic Center 9am 12 noon Problem Solving Strategy 1 Calculate torque about appropriate point S S 2 Calculate angular momentum about S L S 3 Apply approximation that to decide which contribution to the angular momentum about P is changing in time Calculate dL S dt 4 Apply torque law P dL S dt to determine direction and magnitude of precession angular velocity 5 Apply Newton s Second Law to center of mass motion Review Table Problem Gyroscope Consider a flywheel that is spinning with angular speed s as shown in the figure a What is the precessional angular speed b Does the gyroscope rotate clockwise or counterclockwise about the vertical axis as seen from above Table Prob Gyroscope on Rotating Platform A gyroscope consists of an axle of negligible mass and a disk of mass M and radius R mounted on a platform that rotates with angular speed as shown in the figure below The gyroscope is spinning with angular speed Forces Fa and Fb act on the gyroscopic mounts The goal of this problem is to find the magnitudes of the forces Fa and Fb You may assume that the moment of inertia of the gyroscope about an axis passing through the center of mass normal to the plane of the disk is given by I n a Calculate the torque about the center of mass of the gyroscope b Calculate the angular momentum about the center of mass of the gyroscope c Use Newton s Second Law find a relationship between Fa and Fb the mass M of the gyroscope and the gravitational constant g d Use the torque equation and Newton s Second Law to find expressions for Fa and Fb Table Problem Mill Stone In a mill grain is ground by a massive wheel that rolls without slipping in a circle on a flat horizontal mill stone driven by a vertical shaft The rolling wheel has mass M radius b and is constrained to roll in a horizontal circle of radius R at angular speed The wheel pushes down on the lower mill stone with a force equal to twice its weight normal force The mass of the axle of the wheel can be neglected Express your answers to the following questions in terms of R b M and g as needed The goal of this problem is to find a What is the relation between the angular speed of the wheel about its axle and the angular speed about the vertical axis b Find the time derivative of the angular momentum about the joint about the point P in the figure above dL P dt c What is the torque about the joint about the point P in the figure above d What is the value of Worked Example Sopwith Camel The Sopwith Camel was a single engine fighter plane flown by British pilots during WWI and also by the character Snoopy in the Peanuts comic strip It was powered by a radial engine and the entire engine rotated with the propeller The Camel had an unfortunate property if the pilot turned to the right the plane Tended to go into dive while a left turn caused the plane to climb steeply These tendencies caused inexperienced pilots to crash or stall during takeoff Discussion Sopwith Camel From the perspective of the pilot who sat behind the engine did the engine rotate clockwise or counter clockwise Odd tables Argue on the basis of torque and third law pairs Even tables Argue on the basis of conservation of angular momentum in the horizontal plane Worked Example Sopwith Camel Torque Assume one of the two possible directions of rotation and see if it gives the correct result if cw L right turn dL dt on engine on plane results in nose pitching down so clockwise is correct Worked Example Sopwith Camel Angular momentum The pilot begins turning right using the rudder on the tail section applying an external torque changing Lz The large horizontal component of the engine s L swings right If the pilot does not use the elevators to apply another external torque the horizontal component of the plane s motion must counter the engine to ensure no net change in horizontal angular momentum if cw L right turn dL engine dL plane obtained by rotating plane nose downward Concept Question Stabilizing a Turning Car When making a turn every car has a tendency to roll over because its center of mass is above the plane where the wheels contact the road Imagine a race car going counter clockwise on a circular track It could mitigate this effect by mounting a gyroscope on the car To be effective the angular velocity vector of the gyro should point 1 ahead 2 behind 3 to the left 4 to the right 5 up 6 down Table Problem Stabilizing a Car When an automobile rounds a curve at high speed in the figure below the car is turning left the loading weight distribution on the wheels is markedly changed For sufficiently high speeds the loading on the inside wheel goes to zero at which point the car starts to roll over The tendency can be avoided by mounting a large spinning flywheel on the car a What should be the sense of rotation of the flywheel to help equalize the loading Be sure that your method works for cars turning in either direction b Show that for a disk shaped flywheel of mass mw and radius R the requirement for equal loading is that the angular speed of the flywheel s is related to the speed of the car vcm by s 2vcm mc h mw R 2 where mc is the mass of the car and flywheel and h is the height of the center of mass of the car including the flywheel above the road Assume the road is unbanked Appendix Why a Gyroscope Precesses Deflection of Particle by Small Impulse If the impulse I p1 the primary effect is to rotate p about the x axis by a small angle Note the symbol I denotes the magnitude of the impulse and is not the moment of inertia Deflection of Particle by Small Impulse I p Fave t L ave t r Fave t L r Fave t L r I The application of I causes a change in the angular momentum L through the torque equation Deflection of Particle by Small Impulse As a result L rotates about the x axis by a small angle Note that although I is in the z direction L is in the negative y direction Effect of Small Impulse on Tethered Ball The ball is attached to a string rotating about a fixed point Neglect gravity Effect of Small Impulse on Tethered Ball The ball is …
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