Fall 2012 PHYS 172: Modern Mechanics Lecture 15 – Multiparticle Systems Chapter 9.1 – 9.2 EVENING EXAM II 8:00-9:30 PM TUES. OCT 23 Room 112 Covers through Chapter 8 and through Lecture 14, inclusive You may bring one sheet of paper, two sides, with equations, notes, etc. to use during the exam

Clicker question: (Person is jumping upward) Forces on person: normal floor force and gravity : Motion of a multiparticle system A) B) C) g

Clicker question: This and rest of clicker Q’s for discussion only Floor pushes jumper up: does it do any work? A)It does B)It does not Energy

Clicker question: This and rest of clicker Q’s for discussion only Floor pushes jumper up: does it do any work? No motion of floor, so it cannot do any work. Internal forces (muscles) speed up the upper body, and therefore speed up the center of Mass of the person. Energy

The Momentum Principle: center of mass Net momentum, nonrelativistic case: Center of mass: For multiparticle system: Total momentum

Center of mass of several large objects

The Momentum Principle for multiparticle system for v << c r cm

8 Example: ice skater

i>clicker: hockey pucks Two hockey pucks are pulled using attached strings as shown using the same force. The string in case B is wrapped around the puck and can unwind freely. In case A it is attached to the center. Which hockey puck will accelerate quicker? ABAB A) – puck A B) – puck B C) – the same

Example: hockey pucks d1d1 d2d2 ABAB Which puck will acquire larger kinetic energy? ABAB A) – puck A B) – puck B C) – the same

Example: hockey pucks d1d1 d2d2 ABAB Which puck will acquire larger kinetic energy? Extra kinetic energy of rotation in case B ABAB B) – puck B

Kinetic energy of a multiparticle system Can separate total kinetic energy K tot into different parts: (Derivation: 9.8 in the book) Translational, motion of center of mass Motion of parts relative to center of mass Can separate K rel into two* types of motion: (for a bound system) Vibration Rotation in respect to center of mass

Translational kinetic energy Translational kinetic energy: (motion of center of mass) (nonrelativistic case) Clicker: A system is initially at rest and consists of a man with a bottle sitting on ice (ignore friction). The man then throws the bottle away as shown. The velocity of the center of mass v cm will be: A)Zero B)Directed to right C)Directed to left

Translational kinetic energy Translational kinetic energy: (motion of center of mass) (nonrelativistic case) Clicker: A system is initially at rest and consists of a man with a bottle sitting on ice (ignore friction). The man then throws a bottle away as shown. The translational kinetic energy of the system will be: A)Zero B)> 0 C)< 0

Translational kinetic energy Translational kinetic energy: (motion of center of mass) (nonrelativistic case) A system is initially at rest and consists of a man with a bottle sitting on ice (ignore friction). The man then throws a bottle away as shown. Exactly what kind of kinetic energy does this system now have? This is an UNBOUND system, and there’s now a THIRD kind of internal energy, free kinetic energy of constituent parts RELATIVE TO the center of mass.

Vibrational kinetic energy - Net momentum = 0 - Energy is constant (sum of elastic energy and kinetic energy)

Rotational kinetic energy - Net momentum = 0 - Energy is constant Motion around of center of mass

Rotation and vibration CM Rotation and vibration and translation

Gravitational potential energy of a multiparticle system Gravitational energy near the Earth’s surface M y cm

Example: Rotation and translation Assume all mass is in the rim Energy principle: =0 IF WHEEL ROLLS WITHOUT SLIPPING Point of contact is at rest

Rotational kinetic energy A “hoop” has all its mass at the periphery, so its rotational kinetic energy is just ½ M v rel 2 A disc has only half as much K rot because the mass which is closer to the center doesn’t move as fast. A sphere has only 40% as much K rot as a hoop, since even more of the sphere’s mass is close to the axis of rotation, compared to a disc. For the hoop rolling downhill, K rot is equal to K trans and this slows down the translational motion the most, amongst these three shapes. Sliding with no rotation would let ALL of U appear as K trans DEMONSTRATION

Rotational kinetic energy An extreme example is a yo-yo, where the axle is tiny and as the string unwinds, the periphery of the yo-yo spins much faster than the rate of drop of the CM of the yo-yo. Result: most of the KE of the yo-yo is ROTATIONAL, and very little is in K trans, hence, as the yo-yo loses Gravitational U, it drops very slowly. And at the bottom, the “sleeping” yo-yo is effectively a flywheel, storing K which can be converted back to U as the yo-yo climbs back up the string. (for the yo-yo to be able to “sleep”, the string has a loop at the bottom, rather than being attached to the axle)