PHYS 172 Modern Mechanics Lecture 4 Summer 2011

Predictions using the Momentum Principle The Momentum Principle Update form of the momentum principle

Physical models Not what we mean by “Ideal body” “Spherical cow” Ideal model: ignore

Example: colliding students Two students are late for class and run into each other

y Example: colliding students Strategy: x

Example: colliding students y Strategy: x What is the collision time? What is the

The Four Fundamental Forces “Composite” forces like the spring force, air drag, friction, etc.

Newton’s Great Insight: The force that attracts things toward the earth (e. g. a

The gravitational force law m 2 Newton Cavendish Gravitational constant m 1 m 2

Predicting motion of a planet Where will the planet be after one month? Use

Iterative prediction of a motion of one planet Simple case: one planet star is

Iterative prediction of motion Real case: many objects are free to move 1. Calculate

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PHYS 172: Modern Mechanics Lecture 4 Summer 2011 Read 2. 4 -2. 8

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

Physical models Not what we mean by “Ideal body” “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, ground, air, second spherical student Force: Gravity, Normal, Friction, Air Resistance, Other Student – unknown!

y Example: colliding students Strategy: x

Example: colliding students y Strategy: x What is the collision time? What is the initial momentum? Find F: Assume: vi =5 m/s, x=0. 05 m Assume: m=60 kg

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 m 2 Newton Cavendish Gravitational constant m 1 m 2 m 1

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: 2. Update momentum Choose t short enough (F & v do not change much) 3. Calculate v and update position 4. Repeat Critical parameter: t

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