Chapter 8 Rotational Motion and Equilibrium 8.1 Rigid Bodies, Translations, and Rotations A rigid body is an object or a system of particles in which the distances between particles are fixed (remain constant). In other words, a rigid body must be solid (but not all solid bodies are rigid). 1 2 8.1 Rigid Bodies, Translations, and Rotations 8.1 Rigid Bodies, Translations, and Rotations A rigid body may have translational motion, rotational motion, or a combination. It may roll with or without slipping. For an object that is rolling without slipping, 3 4 Question 8.1a Bonnie and Klyde I Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one complete revolution every 2 seconds. Klyde s angular velocity is: a) same as Bonnie s b) twice Bonnie s c) half of Bonnie s d) one-quarter of Bonnie s e) four times Bonnie s Question 8.1b Bonnie and Klyde II Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds. Who has the larger linear (tangential) velocity? a) Klyde b) Bonnie c) both the same d) linear velocity is zero for both of them ω ω Klyde Bonnie Klyde Bonnie 5 6 1
It takes a force to start an object rotating; that force is more effective the farther it is from the axis of rotation, and the closer it is to being perpendicular to the line to that axis. The perpendicular distance from the line of force to the axis of rotation is called the lever arm. The product of the force and the lever arm is called the torque. 7 8 Question 8.4 Using a Wrench You are using a wrench to loosen a rusty nut. Which arrangement will be the most effective in loosening the nut? a b Torque is a vector (it produces an angular acceleration), and its direction is along the axis of rotation, with the sign given by the righthand rule. c d e) all are equally effective 9 10 In order for an object to be in equilibrium, the net force on it must be zero, and the net torque on it must be zero as well. Question 8.5 Two Forces Two forces produce the same a) yes torque. Does it follow that they b) no have the same magnitude? c) depends 11 12 2
The left stick and the triangle are in equilibrium; they will neither translate nor rotate. The stick on the right has no net force on it, so its center of mass will not move; the torque on it is not zero, so it will rotate. For an object to be stable, there must be no net torque on it around any axis. The axis used in calculation may be chosen for convenience when there is no motion. 13 14 If an object is in stable equilibrium, any displacement from the equilibrium position will create a torque that tends to restore the object to equilibrium. Otherwise the equilibrium is unstable. Whether equilibrium is stable or unstable depends on the width of the base of support. 15 16 Question 8.6 Closing a Door In which of the cases shown below a) F 1 is the torque provided by the b) F 3 applied force about the rotation c) F 4 axis biggest? For all cases the d) all of them magnitude of the applied force is the same. e) none of them Question 8.13 Balancing Rod a) ¼ kg A 1-kg ball is hung at the end of a rod 1-m long. If the system balances at a b) ½ kg point on the rod 0.25 m from the end c) 1 kg holding the mass, what is the mass of d) 2 kg the rod? e) 4 kg 1m 1kg 17 18 3
The net torque on an object causes its angular acceleration. For a point particle, the relationship between the torque, the force, and the angular acceleration is relatively simple. We can consider an extended object to be a lot of near-point objects stuck together. Then the net torque is: The quantity inside the parentheses is called the moment of inertia, I. 19 20 The moments of inertia of certain symmetrical shapes can be calculated. Here is a sample: The parallel-axis theorem relates the moment of inertia about any axis to the moment of inertia about a parallel axis that goes through the center of mass. 21 22 8.4 Rotational Work and Kinetic Energy The work done by a torque: As usual, the power is the rate at which work is done: 8.4 Rotational Work and Kinetic Energy The work energy theorem still holds the net work done is equal to the change in the kinetic energy. This gives us the form of the rotational kinetic energy. 23 24 4
8.4 Rotational Work and Kinetic Energy There is a strict analogy between linear and rotational dynamic quantities. Definition of angular momentum: the product of a moment arm( r) and a linear momentum (p). In vector form (the direction is again given by the right-hand rule): 25 26 The rate of change of the angular momentum is the net torque: Internal forces can change a system s moment of inertia; its angular speed will change as well. Angular momentum is conserved: In the absence of an external, unbalanced torque, the total (vector) angular momentum of a system is conserved (remains constant). 27 28 This can be demonstrated in the classroom. (What purpose do the hand weights serve?) The conservation of angular momentum means its direction cannot change in the absence of an external torque. This gives spinning objects remarkable stability. 29 30 5
An external torque on a rotating object causes it to precession. Question 8.10 Figure Skater A figure skater spins with her arms extended. When she pulls in her arms, she reduces her rotational inertia and spins faster so that her angular momentum is conserved. Compared to her initial rotational kinetic energy, her rotational kinetic energy after she pulls in her arms must be a) the same b) larger because she s rotating faster c) smaller because her rotational inertia is smaller 31 32 Question 8.11 Two Disks Two different spinning disks have the same angular momentum, but disk 1 has more kinetic energy than disk 2. a) disk 1 b) disk 2 c) not enough info Which one has the bigger moment of inertia? L L Disk 1 Disk 2 33 6