29.1 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.
39.1 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
49.1 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.
59.1 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)
69.1 The Action of Forces and Torques on Rigid Objects Example 2 The Achilles TendonThe tendon exerts a force of magnitude790 N. Determine the torque (magnitudeand direction) of this force about theankle joint.
79.1 The Action of Forces and Torques on Rigid Objects
89.2 Rigid Objects in Equilibrium If a rigid body is in equilibrium, neither its linear motion nor itsrotational motion changes.
99.2 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.
109.2 Rigid Objects in Equilibrium Reasoning StrategySelect the object to which the equations for equilibrium are to be applied.2. Draw a free-body diagram that shows all of the external forces acting on theobject.Choose a convenient set of x, y axes and resolve all forces into componentsthat lie along these axes.Apply the equations that specify the balance of forces at equilibrium. (Set thenet force in the x and y directions equal to zero.)Select a convenient axis of rotation. Set the sum of the torques about thisaxis equal to zero.6. Solve the equations for the desired unknown quantities.
119.2 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.
17Example 6 The Center of Gravity of an Arm The horizontal arm is composedof three parts: the upper arm (17 N),the lower arm (11 N), and the hand(4.2 N).Find the center of gravity of thearm relative to the shoulder joint.
19Conceptual Example 7 Overloading a Cargo Plane 9.3 Center of GravityConceptual Example 7 Overloading a Cargo PlaneThis accident occurred because the plane was overloaded towardthe rear. How did a shift in the center of gravity of the plane causethe accident?
209.4 Newton’s Second Law for Rotational Motion About a Fixed Axis Moment of Inertia, I
219.4 Newton’s Second Law for Rotational Motion About a Fixed Axis Net externaltorqueMoment ofinertia
229.4 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.
239.4 Newton’s Second Law for Rotational Motion About a Fixed Axis Example 9 The Moment of Inertial 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.
249.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
259.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
269.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
279.4 Newton’s Second Law for Rotational Motion About a Fixed Axis Example 12 Hoisting a CrateThe combined moment of inertia of the dual pulley is 50.0 kg·m2. Thecrate weighs 4420 N. A tension of 2150 N is maintained in the cableattached to the motor. Find the angular acceleration of the dualpulley.
289.4 Newton’s Second Law for Rotational Motion About a Fixed Axis equal
299.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
319.5 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)
339.5 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)
349.5 Rotational Work and Energy Example 13 Rolling CylindersA thin-walled hollow cylinder (mass = mh, radius = rh) anda solid cylinder (mass = ms, radius = rs) start from rest atthe top of an incline.Determine which cylinderhas the greatest translationalspeed upon reaching thebottom.
359.5 Rotational Work and Energy ENERGY CONSERVATION
369.5 Rotational Work and Energy The cylinder with the smaller momentof inertia will have a greater final translationalspeed.
37DEFINITION 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
38PRINCIPLE OF CONSERVATION OFANGULAR MOMENTUM The angular momentum of a system remains constant (isconserved) if the net external torque acting on the systemis zero.