2 9.1 The Action of Forces and Torques on Rigid Objects In pure translational motion, all points on anobject travel on parallel paths.The most general motion is a combination oftranslation and rotation.
3 9.1 The Action of Forces and Torques on Rigid Objects According to Newton’s second law, a net force causes anobject to have an acceleration.What causes an object to have an angular acceleration?TORQUE
4 9.1 The Action of Forces and Torques on Rigid Objects The amount of torque depends on where and in what direction theforce is applied, as well as the location of the axis of rotation.
5 9.1 The Action of Forces and Torques on Rigid Objects DEFINITION OF TORQUEMagnitude of Torque = (Magnitude of the force) x (Lever arm)Direction: The torque is positive when the force tends to produce acounterclockwise rotation about the axis.SI Unit of Torque: newton x meter (N·m)
6 9.2 Rigid Objects in Equilibrium If a rigid body is in equilibrium, neither its translational motion nor itsrotational motion changes.
7 9.2 Rigid Objects in Equilibrium EQUILIBRIUM OF A RIGID BODYA rigid body is in equilibrium if it has zero translationalacceleration and zero angular acceleration. In equilibrium,the sum of the externally applied forces is zero, and thesum of the externally applied torques is zero.
8 9.2 Rigid Objects in Equilibrium Example 3 A Diving BoardA woman whose weight is 530 N ispoised at the right end of a diving boardwith length 3.90 m. The board hasnegligible weight and is supported bya fulcrum 1.40 m away from the leftend.Find the forces that the bolt and thefulcrum exert on the board.
13 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis Moment of Inertia, I
14 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis Net externaltorqueMoment ofinertia
15 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis ROTATIONAL ANALOG OF NEWTON’S SECOND LAW FORA RIGID BODY ROTATING ABOUT A FIXED AXISRequirement: Angular accelerationmust be expressed in radians/s2.
16 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis Example 9 The Moment of Inertia Depends on Wherethe Axis Is.Two particles each have mass and are fixed at theends of a thin rigid rod. The length of the rod is L.Find the moment of inertia when this objectrotates relative to an axis that isperpendicular to the rod at(a) one end and (b) the center.
17 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
18 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
19 9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
21 9.5 Rotational Work and Energy DEFINITION OF ROTATIONAL WORKThe rotational work done by a constant torque inturning an object through an angle isRequirement: The angle mustbe expressed in radians.SI Unit of Rotational Work: joule (J)
23 9.5 Rotational Work and Energy DEFINITION OF ROTATIONAL KINETIC ENERGYThe rotational kinetic energy of a rigid rotating object isRequirement: The angular speed mustbe expressed in rad/s.SI Unit of Rotational Kinetic Energy: joule (J)
24 DEFINITION OF ANGULAR MOMENTUM The angular momentum L of a body rotating about afixed axis is the product of the body’s moment ofinertia and its angular velocity with respect to thataxis:Requirement: The angular speed mustbe expressed in rad/s.SI Unit of Angular Momentum: kg·m2/s
25 PRINCIPLE OF CONSERVATION OF ANGULAR MOMENTUM The angular momentum of a system remains constant (isconserved) if the net external torque acting on the systemis zero.
26 Conceptual Example 14 A Spinning Skater 9.6 Angular MomentumConceptual Example 14 A Spinning SkaterAn ice skater is spinning with botharms and a leg outstretched. Shepulls her arms and leg inward andher spinning motion changesdramatically.Use the principle of conservationof angular momentum to explainhow and why her spinning motionchanges.
27 Problems to be solved9.6, 9.12, 9.14, 9.21, 9.25, 9.40, 9.51, 9.61, 9.69, 9.74B9.1 A bicycle wheel has a mass of 2kg and a radius of 0.35m. What is its moment of inertia? Ans: 0.245kgm2
28 B9. 2 A grinding wheel, a disk of uniform thickness, has a radius of 0 B9.2 A grinding wheel, a disk of uniform thickness, has a radius of 0.08m and a mass of 2kg. (a) What is its moment of inertia? (b) How large a torque is needed to accelerate it from rest to 120rad/s in 8s? Ans: (a) kgm2 (b) 0.096Nm
29 B9.3 A student holding a rod by the centre subjects it to a torque of 1.4Nm about an axis perpendicular to the rod, turning it through 1.3rad in 0.75s. When the student holds the rod by the end applies the same torque to the rod, through how many radians will the rod turn in 1.0s?Ans: 0.58rad
30 B9. 4 An ice skater starts spinning at a rate of 1 B9.4 An ice skater starts spinning at a rate of 1.5rev/s with arms extended. She then pulls her arms in close to her body, resulting in a decrease of her moment of inertia to three quarters of the initial value. What is the skater’s final angular velocity? Ans: 2rev/s