PHYS 172: Modern Mechanics Summer 2011 Lecture 4 Read 2.4-2.8

Predictions using the Momentum Principle Update form of the momentum principle 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: No numbers – a problem from real world. One needs to guess best parameters and build a model System: one spherical student Surroundings: earth, ground, air, second spherical student Force: Gravity, Normal, Friction, Air Resistance, Other Student – unknown!

Example: colliding students y Strategy:

Example: colliding students y Strategy: What is the collision time? Assume: vi =5 m/s, x=0.05m 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 m2 Cavendish Gravitational constant Newton m1 m1 m2 Newton: ~m1m2/r^2, Cavendish in 1798 – measured constant G In physics, the Cavendish experiment was the first experiment to accurately measure the gravitational constant by measuring the force of gravity between two masses in the laboratory. The experiment was originally proposed by John Michell, who constructed a torsion balance apparatus, but Michell died without completing the experiment. After his death in 1793 the apparatus passed to Francis John Hyde Wollaston, who gave it to Henry Cavendish. Cavendish rebuilt the apparatus, staying close to Michell's plan. Cavendish carried out a series of careful experiments reported in the Philosophical Transactions in 1798. The apparatus comprised a six-foot (1.8 m) wooden rod with metal spheres attached to each end, suspended from a wire. Two 350 pound (159 kg) lead spheres placed nearby exerted just enough gravitational force to tug at the end-weights, causing the wire to twist. To prevent air currents from interfering, Cavendish set up the apparatus in a wind-proof room and measured the twist (torsion) of the wire using a telescope. From the twisting force in the wire and the known masses of the spheres, Cavendish was able to calculate the value of the gravitational constant. Since the force of the gravitational attraction of the earth for an object of known mass could be measured directly, the measurement of the gravitational constant allowed the mass of the earth to be calculated for the first time. This in turn allowed the calculation of the masses of the sun, the moon, and the other planets. Modern measurements of the gravitational constant still use variations of this method. A description of Cavendish's experiment and a summary of several similar experiments are given by the 1911 Encyclopædia Britannica 9

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. Same strategy – predicting trajectory of a rocket 10

Iterative prediction of a motion of one planet Simple case: one planet star is fixed in space Calculate gravitational force: 2. Update momentum Choose t short enough (F & v do not change much) 3. Calculate v and update position Same strategy – predicting trajectory of a rocket 4. Repeat Critical parameter: t 11

Iterative prediction of motion Real case: many objects objects are free to move 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! 12