2008 Physics 2111 Fundamentals of Physics Chapter 11 1 Fundamentals of Physics Chapter 12 Rolling, Torque & Angular Momentum 1.Rolling 2.The Kinetic Energy of Rolling Rolling as Pure Rotation 3.The Forces of Rolling Friction & Rolling Rolling Down a Ramp 4.The Yoyo 5.Torque Revisited 6.Angular Momentum 7.Newton’s 2 nd Law in Angular Momentum Form 8.Angular Momentum of a System of Particles 9.Angular Momentum of a Rigid Body Rotating About a Fixed Axis 10.Conservation of Angular Momentum Review & Summary Questions Exercises & Problems

2008 Physics 2111 Fundamentals of Physics Chapter 11 2 Rolling without Slipping Units: (m/s) (rad/s) (m) Units: (m) (radians) (m)

2008 Physics 2111 Fundamentals of Physics Chapter 11 3 Rolling

2008 Physics 2111 Fundamentals of Physics Chapter 11 4 Example: Value of x for rolling without slipping? Rolling without slipping: x > 2/5 R  topspin x = 0  no initial spin Independent of m and F!

2008 Physics 2111 Fundamentals of Physics Chapter 11 5 The Forces of Rolling No Initial Rotation:Topspin: Frictional force reduces the speed & increases the angular speed until: Frictional force accelerates the ball in the direction of motion: increasing v decreasing  Rolling with slipping initially Rolling without slipping

2008 Physics 2111 Fundamentals of Physics Chapter 11 6 Translation & Rotation Add: Linear velocity due to rotation Linear velocity of the center-of-mass pure rotation + pure translation = rolling motion To get: Linear velocity of each point on a rolling wheel. See p247 in HRW.

2008 Physics 2111 Fundamentals of Physics Chapter 11 7 The Kinetic Energy of Rolling The instantaneous axis of rotation is at Point P: A rolling object has two types of kinetic energy: a rotational KE due to its rotation about its center of mass and a translational KE due to the translation of its center of mass.

2008 Physics 2111 Fundamentals of Physics Chapter 11 8 The Forces of Rolling Friction & Rolling: friction is required! No slipping (aka “smooth rolling”) f s acts on the wheel at P, opposing its tending to slide. Wheel is rotating about point P. No work is done by f s ! The wheel does not slow down.

2008 Physics 2111 Fundamentals of Physics Chapter 11 9 Rolling down a Ramp  > 0 counter clockwise a < 0 down the ramp Any body rolling down a ramp!

2008 Physics 2111 Fundamentals of Physics Chapter Uniform Ball Rolling Down Conservation of energy: M = 6.00 kg Radius R  = 30 0 h = 1.20 m v = ? f s = ?

2008 Physics 2111 Fundamentals of Physics Chapter Hoop, Disk and Sphere rolling down a ramp:

2008 Physics 2111 Fundamentals of Physics Chapter Wheel has constant acceleration & no slipping f S = ? I = ?

2008 Physics 2111 Fundamentals of Physics Chapter Ball rolls down the hill x x = ? Energy: Projectile: H = 6.0 m h = 2.0 m

2008 Physics 2111 Fundamentals of Physics Chapter Loop the Loop Energy: Newton:

2008 Physics 2111 Fundamentals of Physics Chapter Torque “Revisited”

2008 Physics 2111 Fundamentals of Physics Chapter Angular Momentum

2008 Physics 2111 Fundamentals of Physics Chapter All of the particles have the same momentum. Largest Magnitude of Angular Momentum about O ??? Negative Angular Momentum??? Zero Angular Momentum??? 1 and 3 2 and 3 5

2008 Physics 2111 Fundamentals of Physics Chapter Angular Momentum b) Torque = ? a)Angular momentum = ? out of the page m = 2.0 kg r = 3.0 m v = 4.0 m/s F = 2.0 N

2008 Physics 2111 Fundamentals of Physics Chapter Newton’s 2 nd Law in Angular Momentum Form The vector sum of all the torques acting on a particle is equal to the time rate of change of the angular momentum of that particle. Time derivative of the angular momentum:

2008 Physics 2111 Fundamentals of Physics Chapter Checkpoint Largest Torque about Point O??? Positive Torque about Point O??? Zero Torque about Point O???

2008 Physics 2111 Fundamentals of Physics Chapter Angular Momentum of a System of Particles The net external torque acting on a system of particles is equal to the time rate of change of the system’s total angular momentum. Internal torques sum to zero:

2008 Physics 2111 Fundamentals of Physics Chapter Angular Momentum About a Fixed Axis

2008 Physics 2111 Fundamentals of Physics Chapter A Rigid Body Rotating About a Fixed Axis  Angular momentum of a mass element wrt the origin: Component of angular momentum in z-direction:

2008 Physics 2111 Fundamentals of Physics Chapter A Rigid Body Rotating About a Fixed Axis  Total angular momentum in z-direction:

2008 Physics 2111 Fundamentals of Physics Chapter Checkpoint : same radius same mass same force Largest Angular Momentum about Center??? Largest Angular Speed after a time interval??? Newton’s 2 nd Law: All the angular momenta are the same.

2008 Physics 2111 Fundamentals of Physics Chapter Conservation of Angular Momentum The Spinning Volunteer: If the net external torque acting on a system is zero, the angular momentum of the system remains constant, no matter what changes take place within the system. Spins faster!

2008 Physics 2111 Fundamentals of Physics Chapter Conservation of Angular Momentum If the component of the net external torque on a system along a certain axis is zero, then the component of angular momentum of the system along that axis remains constant, no matter what changes take place within the system. The Diver:

2008 Physics 2111 Fundamentals of Physics Chapter The system = stool + person + spinning wheel Conservation of Angular Momentum

2008 Physics 2111 Fundamentals of Physics Chapter A train starts running on a wheel;  = ? A train starts running on a wheel;  = ?

2008 Physics 2111 Fundamentals of Physics Chapter The sun goes out and then shrinks! The sun will become a white dwarf. Sun goes out Shrinks size Earth Period ?