Fall 2011 PHYS 172: Modern Mechanics Lecture 4 – Physical Models, Fundamental Interactions Read 2.7–2.8,

Predictions using the Momentum Principle The Momentum Principle Update form of the momentum principle Short enough, F~const For components:

Example Force: provided by a spring stretched by  L=4 cm interaction duration: 1 s ? Find momentum p f if p i = kg. m/s no friction k S =500 N. m 1. Force: 2. Momentum: NB: force must not change during  t N. s = kg. m/s 2. s = kg. m/s

Physical models “Spherical cow” Ideal model: ignore factors that have no significant effect on the outcome

Example: colliding students Two students are late for class and run into each other head-on. Q: Estimate the force that one student exerts on the other during collision Simplest model: System:one spherical student Surroundings: earth, floor, air, second spherical student Force:Earth, floor, air, other student – unknown!

Example: colliding students Strategy: x y

Example: colliding students Strategy: x y Assume: v i =5 m/s,  x =0.05m What is the collision time? What is the initial momentum?Assume: m=60 kg Find F:

The Four Fundamental Forces “Composite” forces like the spring force, air drag, friction, etc. are combinations of these four fundamental forces

Newton’s Great Insight: The force that attracts things toward the earth (e.g. a falling apple) is the same force that keeps planets orbiting about the sun

The gravitational force law m1m1 m2m2 Newton m1m1 m2m2 Cavendish Gravitational constant

Predicting motion of a planet Where will the planet be after one month? Use position update formula: If we assume that velocity is constant Does not work because the force is changing the velocity! The force changes with position. The momentum changes with position. In general, there is no algebraic equation to predict motion of more than 2 interacting objects.

Iterative prediction of a motion of one planet Simple case: one planet star is fixed in space 1.Calculate gravitational force: 3. Calculate v and update position 2. Update momentum Choose  t short enough (F & p do not change much) 4. Repeat Critical parameter:  t

Iterative prediction of motion Real case: many objects objects are free to move 1.Calculate net force on each mass: 3. Calculate v and update position of each mass 2. Update momentum of each mass Choose  t short enough (F & v do not change much) 4. Repeat  t is a critical parameter! Iterative approach: works for any kind of force, not just gravity!

Iterative prediction of motion: throwing a rock Constant Force (mg) + Momentum Principle = Projectile Motion (curved path) But, can also add air resistance = non-constant force