Rotational Kinematics Chapter 8 Rotational Kinematics

8.1 Rotational Motion and Angular Displacement The angle through which the object rotates is called the angular displacement.

8.1 Rotational Motion and Angular Displacement DEFINITION OF ANGULAR DISPLACEMENT When a rigid body rotates about a fixed axis, the angular displacement is the angle swept out by a line passing through any point on the body and intersecting the axis of rotation perpendicularly. By convention, the angular displacement is positive if it is counterclockwise and negative if it is clockwise. SI Unit of Angular Displacement: radian (rad)

8.1 Rotational Motion and Angular Displacement For a full revolution:

8.2 Angular Velocity and Angular Acceleration

8.2 Angular Velocity and Angular Acceleration Changing angular velocity means that an angular acceleration is occurring. DEFINITION OF ANGULAR ACCELERATION SI Unit of Angular acceleration: radian per second squared (rad/s2)

8.2 Angular Velocity and Angular Acceleration Example 4 A Jet Revving Its Engines As seen from the front of the engine, the fan blades are rotating with an angular speed of -110 rad/s. As the plane takes off, the angular velocity of the blades reaches -330 rad/s in a time of 14 s. Find the angular acceleration, assuming it to be constant.

8.2 Angular Velocity and Angular Acceleration

8.3 The Equations of Rotational Kinematics Recall the equations of kinematics for constant acceleration. Five kinematic variables: 1. displacement, x 2. acceleration (constant), a 3. final velocity (at time t), v 4. initial velocity, vo 5. elapsed time, t

8.3 The Equations of Rotational Kinematics The equations of rotational kinematics for constant angular acceleration: ANGULAR ACCELERATION ANGULAR VELOCITY TIME ANGULAR DISPLACEMENT

8.3 The Equations of Rotational Kinematics

8.4 Angular Variables and Tangential Variables

8.4 Angular Variables and Tangential Variables

8.4 Angular Variables and Tangential Variables

8.4 Angular Variables and Tangential Variables Example 6 A Helicopter Blade A helicopter blade has an angular speed of 6.50 rev/s and an angular acceleration of 1.30 rev/s2. For point 1 on the blade, find the magnitude of (a) the tangential speed and (b) the tangential acceleration.

8.4 Angular Variables and Tangential Variables 1

8.4 Angular Variables and Tangential Variables 1

8.4 Angular Variables and Tangential Variables 1 2

8.5 Centripetal Acceleration and Tangential Acceleration always

8.5 Centripetal Acceleration and Tangential Acceleration Example 7 A Discus Thrower Starting from rest, the thrower accelerates the discus to a final angular speed of +15.0 rad/s in a time of 0.270 s before releasing it. During the acceleration, the discus moves in a circular arc of radius 0.810 m. The angular acceleration is constant. Find the magnitude of the total acceleration.

8.5 Centripetal Acceleration and Tangential Acceleration

8.5 Centripetal Acceleration and Tangential Acceleration if α is constant: aT is constant, ac is not! not constant constant

8.5 Centripetal Acceleration and Tangential Acceleration if α is constant: aT is constant, ac is not! not constant constant

8.5 Centripetal Acceleration and Tangential Acceleration if α is constant: aT is constant, ac is not! not constant constant Φ=tan-1(aT/ac)

8.6 Rolling Motion Translation + rotation

8.6 Rolling Motion Translation + rotation v=va +ωR v=va -ωR va v = - ωR v = +ωR ω R

condition to roll without slipping 8.6 Rolling Motion Translation + rotation v=va +ωR v=va -ωR va v = - ωR v = +ωR ω R v=va=ωR if va= ωR vcontact-point=0 condition to roll without slipping contact point: v=0

condition to roll without slipping 8.6 Rolling Motion Translation + rotation v=va +ωR v=va -ωR v = - ωR v = +ωR vc v=0 v=va=ωR if va= ωR vcontact-point=0 condition to roll without slipping

8.6 Rolling Motion conditions for no slipping speed of the car = speed of axes of wheels acceleration of the car = acc. of axes of wheels

Example 8 An Accelerating Car Starting from rest, the car accelerates 8.6 Rolling Motion Example 8 An Accelerating Car Starting from rest, the car accelerates (without slipping) for 20.0 s with a constant linear acceleration of 0.800 m/s2. The radius of the tires is 0.330 m. What is the angle through which each wheel has rotated?

8.6 Rolling Motion θ α ω ωo t ? -2.42 rad/s2 0 rad/s 20.0 s

9.1 The Action of Forces and Torques on Rigid Objects According to Newton’s second law, a net force causes an object to have an acceleration. What causes an object to have an angular acceleration? TORQUE

9.1 The Action of Forces and Torques on Rigid Objects The amount of torque depends on where and in what direction the force is applied, as well as the location of the axis of rotation.

9.1 The Action of Forces and Torques on Rigid Objects θ DEFINITION OF TORQUE Magnitude of Torque = (Magnitude of the force) x (Lever arm) angle between F and d Direction: The torque is positive when the force tends to produce a counterclockwise rotation about the axis. SI Unit of Torque: newton x meter (N·m)

9.1 The Action of Forces and Torques on Rigid Objects θ angle between F and d

9.2 Rigid Objects in Equilibrium If a rigid body is in equilibrium, neither its linear motion nor its rotational motion changes.

9.2 Rigid Objects in Equilibrium EQUILIBRIUM OF A RIGID BODY A rigid body is in equilibrium if it has zero translational acceleration and zero angular acceleration. In equilibrium, the sum of the externally applied forces is zero, and the sum of the externally applied torques is zero.

9.2 Rigid Objects in Equilibrium Reasoning Strategy Select 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 the object. Choose a convenient set of x, y axes and resolve all forces into components that lie along these axes. Apply the equations that specify the balance of forces at equilibrium. (Set the net force in the x and y directions equal to zero.) Select a convenient axis of rotation. Set the sum of the torques about this axis equal to zero. 6. Solve the equations for the desired unknown quantities.

9.2 Rigid Objects in Equilibrium Torque can be + or - Negative torque: would make a clockwise rotation Positive torque: would make a counterclockwise rotation x x x x x x

9.2 Rigid Objects in Equilibrium Example 3 A Diving Board A woman whose weight is 530 N is poised at the right end of a diving board with length 3.90 m. The board has negligible weight and is supported by a fulcrum 1.40 m away from the left end. Find the forces that the bolt and the fulcrum exert on the board.

9.2 Rigid Objects in Equilibrium

9.2 Rigid Objects in Equilibrium

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis Moment of Inertia, I

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis ROTATIONAL ANALOG OF NEWTON’S SECOND LAW FOR A RIGID BODY ROTATING ABOUT A FIXED AXIS Requirement: Angular acceleration must be expressed in radians/s2.

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis Example 9 The Moment of Inertial Depends on Where the Axis Is. Two particles each have mass and are fixed at the ends of a thin rigid rod. The length of the rod is L. Find the moment of inertia when this object rotates relative to an axis that is perpendicular to the rod at (a) one end and (b) the center.

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis

9.5 Rotational Work and Energy DEFINITION OF ROTATIONAL KINETIC ENERGY The rotational kinetic energy of a rigid rotating object is Requirement: The angular speed must be expressed in rad/s. SI Unit of Rotational Kinetic Energy: joule (J)

9.5 Rotational Work and Energy Example 13 Rolling Cylinders A thin-walled hollow cylinder (mass = mh, radius = rh) and a solid cylinder (mass = ms, radius = rs) start from rest at the top of an incline. Determine which cylinder has the greatest translational speed upon reaching the bottom.

9.5 Rotational Work and Energy ENERGY CONSERVATION

9.5 Rotational Work and Energy The cylinder with the smaller I/(mr2) ratio will have a greater final translational speed.

DEFINITION OF ANGULAR MOMENTUM The angular momentum L of a body rotating about a fixed axis is the product of the body’s moment of inertia and its angular velocity with respect to that axis: Requirement: The angular speed must be expressed in rad/s. SI Unit of Angular Momentum: kg·m2/s

PRINCIPLE OF CONSERVATION OFANGULAR MOMENTUM The angular momentum of a system remains constant (is conserved) if the net external torque acting on the system is zero.

Conceptual Example 14 A Spinning Skater 9.6 Angular Momentum Conceptual Example 14 A Spinning Skater An ice skater is spinning with both arms and a leg outstretched. She pulls her arms and leg inward and her spinning motion changes dramatically. Use the principle of conservation of angular momentum to explain how and why her spinning motion changes.