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Three Dimensional Roation: Problem Solving 8.01 W14D2 Today’s Reading Assignment Young and Freedman: 10.7 Announcements Problem Set 11 Due Thursday Dec 8 9 pm Sunday Tutoring in 26-152 from 1-5 pm 2 Concept Question: Rotating Rod Consider a massless rod of length I with two point-like objects of mass m at each end, rotating about the vertical z-axis with angular speed ω. There is a sleeve on the axis of rotation. At the moment shown in the figure, two forces are acting on the sleeve. The direction of the change of the angular momentum about the center of mass points 1. along the z-axis. 2. along the line formed by the rod. 3. in the plane of the figure, perpendicular to the line formed by the rod. 4. into the plane of the figure. 5. out of the plane of the figure. Concept Q. Answer: Rotating Rod Answer 4. The torque about the center of mass points into the plane of the figure. Therefore the direction of the change of the angular momentum about the center of mass points into the plane of the figure. Problem Solving Strategy r 1. Calculate torque about appropriate point P, P r 2. Calculate angular momentum about P, L P 3. Apply approximation that = to decide which contribution to the angular momentum about P is r changing in time. Calculate dL P / dt r r 4. Apply torque law P dL P / dt to determine direction and magnitude of angular precessional velocity 5. Apply Newton’s Second Law to center of mass motion Table Problem: Suspended Gyroscope A gyroscope wheel is at one end of an axle of length d . The oth er 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 an angle with the vertical. The wheel has mass M and moment of inertia about its center of mass I cm . Its sp in angular speed is . Neglect the mass of the shaft and the mass of the string. Assume . What is the direction and magnitude of precessional angular velocity? the Table Prob. Gyroscope on Rotating Platform A gyroscope c onsists of an axle of negligible mass and a disk of mass M and rad ius R mounted on a platform that ro tates 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 a n axis passing thr ough the center o f mass normal to the plane of the d isk 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 NewtonХs Second Law find a relationship between Fa and Fb , the mass M of the g yroscope, and the gravitational constant g . d) Use the to rque equation and Ne wtonХs Second Law to find expressions for Fa and Fb . Table Problem: Mill Stone In a mill, grain is ground by a massive wheel that ro lls without slipping in a circle on a flat horizontal mill stone driven by a vert ical shaft. The rolli ng wheel has mass M , radius b and is constra ined to roll in a horizonta l circle of radius R at angular speed . The wheel pushes do wn 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 ans wers to the following questions in terms of R , b , M , , and g as need ed. The goal of this problem is to find . a) What is the relation between the angular speed of the whee l about its axle and the angu lar speed about th e vert ical 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 (ab out the point P in the figure above ? d) What is the value of ? Table Problem: 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. Table Problem: 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. 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 Concept Question: Stabilizing a Turning Car Answe r 4. Note that the same orientation of the gyro will stabilize the car when it turns to the right as well. The torque applied to the car by the gyro will change direction, pointing toward the rear of the car instead of the front. But now the result of the curved path is tending to roll the car to the left instead of to the right. The gyro still acts to prevent a roll-over.