Problem Solving 11 Dimensional Rotational Motion and Gyroscopes 8 01 W14D3 Today s Reading Assignment MIT 8 01 Course Notes Chapter 22 Three Dimensional Rotations and Gyroscopes Sections 22 4 Announcements Course Evaluations Forms Available Online at http web mit edu subjectevaluation 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 Rotating Skew Rod Consider a simple rigid body consisting of two particles of mass m separated by a rod of length 2l and negligible mass The midpoint of the rod is attached to a vertical axis that rotates with angular velocity k about the z axis The rod is skewed from the vertical at an angle Set time t 0 when the rod is in the position shown in figure below left At t the rod has rotated to the position shown in the figure below right a Find the direction and magnitude of the angular momentum about the center of mass at t 0 b Find the direction and magnitude of the torque about the center of mass at time t 0 Group Problem Suspended Gyroscope A gyroscope wheel is at one end of an axle of length d The other end of the axle is suspended from a string of length s The wheel is set into motion so that it executes uniform precession in the horizontal plane The string makes a fixed angle with the vertical The wheel has mass M and moment of inertia about its center of mass I cm Its spin angular speed i s Neglect the mass of the shaft and the mass of the string Assume What is the direction and magnitude precession angular velocity of the 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
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