Collisions Momentum: a measure of motion Force: a cause of change in motion What changes when a force is applied? Linear Momentum: p ≡ mv (vector!!!!!) the tendency of an object to pursue straight line motion Impulse: the change in motion “Impulse Power” ?

Conservation of momentum two (or more) bodies + action/reaction + no external forces FAB = - FBA → equal but opposite impulses! → pA + pB = 0 When the net external force on a system is zero, the total momentum of that system is constant. p1 + p2 + p3 + ... is constant Collisions, explosions etc: m1v1 + m2v2 =m1v’1 + m1v’2 Cart Demo's Elastic = kinetic energy is also conserved Inelastic = kinetic energy is lost (some “stickiness”) Completely Inelastic =maximal Kinetic Energy Loss (masses stick together) Superball Demo Application to Bumper Cars

What plays the role of mass, force, momentum, etc? Rotations Rotational Kinematics In close analogy with linear motion with constant acceleration 𝐿𝑖𝑛𝑒𝑎𝑟𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝐴𝑛𝑔𝑢𝑙𝑎𝑟𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑥 θ 𝑣 ω 𝑎 α What plays the role of mass, force, momentum, etc?

Rotational Inertia: like mass from linear motion Rotational Kinetic Energy for a single point particle for a solid rotating object Moment of Inertia I = moment of Inertia = rotational inertia I =  mr2 = m1r12 + m2r22 + m3r32 + ... Rotational Inertia depends upon how the mass is distributed Rotational Inertia: like mass from linear motion F = ma becomes t = Ia where t is the torque (twisting version of force)

L L R2 R a b a b R R

angular momentum (chair) angular momentum (precession) Torque: the rotational analogue of force Torque = force x moment arm  = Fr┴=F r sin  moment arm = perpendicular distance through which the force acts r ┴ r ┴ q=90° q F F r ┴ q F 𝐿𝑖𝑛𝑒𝑎𝑟𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝐴𝑛𝑔𝑢𝑙𝑎𝑟𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑚 𝐼 𝑎 α 𝐹 τ 𝑝 𝐿 τ=𝐼α 𝐾𝐸 𝑟𝑜𝑡 = 1 2 𝐼 ω 2 𝐿=𝐼ω Demo: torque (gyro) angular momentum (chair) angular momentum (precession) Assignment: Tilt-A-Whirl!

Oscillations Equilibrium “restoring force” example: spring, where F = -kx plus Inertia mass → Oscillations Periodic Motion motion that repeats T = period for one full cycle of motion frequency number of cycles per unit time angular frequency w radians per unit time 𝑓= 1 𝑇 ω=2π𝑓= 2π 𝑇

Simple Harmonic Motion of a mass on a spring A is the amplitude of the motion (maximum displacement from equilibrium) motion can be thought of as projection of uniform circular motion SIMULATION! 𝑥=𝐴cos 2π 𝑇 𝑡 =𝐴cos ω𝑡 𝑥=𝑟cos θ 𝑤𝑖𝑡ℎθ=ω𝑡,𝑟=𝐴 𝑣 𝑇 =ω𝑟, 𝑣 𝑥 =− 𝑣 𝑇 sin θ =− 𝑣 𝑇 sin ω𝑡 𝑎 𝑐𝑝 =𝑟 ω 2 , 𝑎 𝑥 =− 𝑎 𝑐𝑝 cos θ =− ω 2 𝑟cos θ 𝑠𝑜 𝑎 𝑥 =− ω 2 𝑥 𝑚𝑎𝑠𝑠−𝑠𝑝𝑟𝑖𝑛𝑔𝑠𝑦𝑠𝑡𝑒𝑚 𝐹=−𝑘𝑥=𝑚𝑎 𝑠𝑜 ω= 𝑘 𝑚 so 𝑓= 1 2π 𝑘 𝑚 Amplitude does not affect frequency!

mass m on a string of length L x s mg T Fnet L The Simple Pendulum mass m on a string of length L This is a small angle approximation Example: How long should a pendulum be in order to have a period of 1.0 s?

Physical Pendulum, center of gravity mg d sin I sin  More SHO variations Physical Pendulum, center of gravity mg d sin I sin  -mg d Ia →  Torsion Pendulum-demo = -  = I  w = ? Pirate Ship Swing Ride Worksheet