Conservation of Momentum Student Notes PDF

Conservation of Momentum Momentum Quantity of motion that an object possesses Linear Momentum Angular Momentum Product of object’s mass and its Product of object’s moment of velocity inertia and its angular velocity...

Conservation of Momentum Momentum Quantity of motion that an object possesses Linear Momentum Angular Momentum Product of object’s mass and its Product of object’s moment of velocity inertia and its angular velocity Units: kg*m2/s Units: kg*m/s Linear Momentum Product of object’s mass and its velocity Race Car Garbage Truck Mass 1496.85 kg 14968.55 kg Velocity 89.41 m/s 11.18 m/s Momentum Linear Momentum How do we change momentum? Linear: Change mass Change velocity Often occurs during collision Garbage Truck Garbage Truck Empty Full Mass 14968.55 kg 22697.62 kg Velocity 11.18 m/s 11.18 m/s Momentu 167348.39 253759.39 m kgm/s kgm/s Linear Momentum Conservation of Momentum Related to Newton’s Law I Newton’s Law 1 = M1=M2 m1v1=m2v 2 Impulse External forces acting on an object change momentum Dependent on Magnitude of force Time it acts 𝐉= 𝐅𝐭 J= Impulse F = Force t = time Impulse Relation b/w impulse and momentum related to Newton’s Law II Impulse Changes in momentum may result from Small F acting for long time Large F acting quickly Examples Hitting a baseball: large force from bat acting very quickly on ball Impulse Can we manipulate impulse? Vertical jump Impulse Can we manipulate impulse? Vertical jump Increase t Not practical b/c force actually as t increases  Must find tradeoff b/w F and t Impulse Can we manipulate impulse? Landing Angular Momentum Quantity of motion that an object possesses Angular Momentum Product of object’s moment of inertia and its angular velocity Top A Top B Mass 0.5 kg 0.5 kg Radius of 0.1 m 0.05 m Rotation Angular 1.5 rad/s 2 rad/s velocity Angular Momentum Angular Momentum How do we manipulate angular momentum? Change I or w To conserve (H stays constant) I = w I = w Angular Momentum Rotating bodies Three axes passing thru COM Transverse: L to R and || to floor Somersault Frontal: front to back and || to floor Tilt Longitudinal: head thru feet Twist Angular Momentum Angular Momentum Example Somersault: How many revolutions can a gymnast perform? Angular Momentum Straight Tucked Can perform 3x as many revolutions tucked (assuming remain tucked throughout) Knowledge Check A figure skater rotates with her arms out to her sides. In this position, her angular momentum is 20 kgm2/s, moment of inertia is 5 kgm2, and her angular velocity is 4 rad/s. If she crosses her arms over her chest, reducing her moment of inertia to 2 kgm2, what is her new angular velocity? a) 5 rad/s b) 10 rad/s c) 1 rad/s d) 40 rad/s Knowledge Check Angular Momentum How do gymnasts initiate somersaults in the air? How do they stop their rotations? Exploit conservation of angular momentum Angular Momentum Start Middle End Position: Position: Position: Tall, upright tucked or straightenin Mechanics: pike g Large I, low w Mechanics: Mechanics: Impression: I, w  I, w traveling up Impression: Impression: not rotating started stopping somersault rotation to in air land Angular Momentum Angular Momentum So how do gymnasts generate twist for aerial skills? Contact Tilt Angular Momentum Contact twist While in contact w/ground Generate moment about longitudinal axis Example: turn shoulder during take-off (Newton’s Law II) Angular Momentum Contact twist Can manipulate rate of twist by applying conservation of angular momentum Pull arms in to I Results in w Angular Momentum Tilt twist When somersaulting if body is tilted away from vertical it will twist Required to keep angular momentum constant about all axes Angular Momentum Tilt twist How can body be tilted? Raising or lowering one arm Turning shoulders when body is piked Extending from pike asymmetrically Angular Momentum Tilt twist Important ONLY works when somersaulting More tilt = faster twist Twist normally can be stopped by reverse process Movements to produce twist are often subtle Can be combined with another method of twist generation to enhance twist Angular Momentum Long jump Jumpers can have 15kgm2/s of angular momentum about transverse axis Goal: land with feet forward not rotated back Have to rotate arms and legs to generate angular momentum Carl Lewis Long Jump  Key Points Impulse and momentum are related through Newton’s 2nd law To manipulate impulse Change F or t… think about which is practical Assume angular momentum is constant Can manipulate I or w Manipulate jump/twist by exploiting conservation of angular momentum OR by generating twist before takeoff Exam 3 Topics Kinetics Mass Newton’s Linear Laws Linear Momentum 1. Inertia Conservation of 2. Acceleration momentum 3. Motion Impulse Ground Reaction Force Angular Momentum Newton’s Angular Laws Contact Twist Whole Body Center of Tilt Twist