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v = rω atan = rα arad = rω2 a a a 2 tan 2 rad Rotation about a Moving Axis • Every motion of of a rigid body can be represented as a combination of motion of the center of mass (translation) and rotation about an axis through the center of mass • The total kinetic energy can always be represented as the sum of a part associated with motion of the center of mass (treated as a point) plus a part asociated with rotation about an axis through the center of mass Total Kinetic Energy Ktotal = (1/2)Mvcm2 + (1/2)Icmω2 CHAPTER 10 DYNAMICS OF ROTATIONAL MOTION A plumbing problem Torque on a pulley Unwinding a winch Rotation about a moving axis Conservation of angular momentum When the sum of the torques of all external forces acting on a system is zero, then THE TOTAL ANGULAR MOMENTUM IS CONSTANT (CONSERVED) The professor as figure skater? •It seems that danger to the instructor is proportional to interest in any given demonstration. Internal forces cannot change the total momentum of a system. Balancing on a teeter-totter – Figure 10.25 • The heavier child must sit closer to balance the torque from the smaller child. •Refer to example 10.11 on page 314. Balanced forces during exercise A gyroscope in the laboratory A 1.31-kg bowling trophy is held at arm’s length, a distance of 0.505 m from the shoulder joint. What torque does the trophy exert about the shoulder if the arm is (a) horizontal, or (b) at an angle of 20° below the horizontal? A school yard teeter-totter with a total length of 5.2 m and a mass of 36 kg is pivoted at its center. A 18 kg child sits on one end of the teeter-totter. (a)Where should a parent push vertically downward with a force of 210 N in order to hold the teetertotter level? (b)Where should the parent push with a force of 310 N? (c)How would your answer to parts (a) and (b) change if the mass of teeter-totter were doubled? Ch 9 Problem 39 The flywheel of a gasoline engine is required to give up 500 J of kinetic energy while its angular velocity decreases from 650 rev/min to 520 rev/min. What moment of inertia is required? Ch 9 Problem 49 A size-5 soccer ball of diameter 22.6 cm and mass 426 g rolls up a hill without slipping, reaching a maximum height of 5.00 m above the base of the hill. We can model this ball as a thin-walled hollow sphere. (a) At what rate was it rotating at the base of the hill? (b) How much rotational kinetic energy did it then have? Ch 9. Problem 51 A solid uniform cylinder and a solid uniform sphere, each with the same mass and diameter, approach a hill rolling with a forward speed of 6.50 m/s. Both of them roll up the hill without slipping. (a) Find the maximum height that each of the centers will reach. (b) Why do they reach different heights? Didn’t both of them have the same speed at the bottom of the hill? (c) Would your answer to part (a) be different if these two objects (1) did not have the same masses or (2) did not have the same diameter? What makes you say this? Ch 10 Problem 27 A certain drawbridge can be modeled as a uniform 15,000 N bar, 12.0 m long, pivoted about its lower end. When this bridge is raised to an angle of 60.0 degrees above the horizontal, the cable holding it suddenly breaks, allowing the bridge to fall. At the instant after the cable breaks, (a) what is this torque on the bridge about the pivot and (b) at what rate is its angular momentum changing?