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First Hour Exam

Prepare for the first hour exam by reviewing topics such as springs, ropes and pulleys, circular motion, non-uniform circular motion, unit vectors in polar coordinates, non-inertial reference frames, resistive forces, and numerical modeling. The exam will consist of approximately 12 multiple choice questions. Don't forget to bring a dark pencil, paper, and a scientific calculator.

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First Hour Exam

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  1. First Hour Exam • Wed, Sep 27, 3:35 pm. • Bring a dark pencil, paper, and a scientific calculator. • There will be approximately 12 multiple choice questions. • Do the problems symbolically, and plug in numbers at the end. You will keep the exam. • I will post a practice exam on the website.

  2. Review from Last Lecture • Springs • F = -kx • Ropes & Pulleys • Tension is the same everywhere in a (light) rope • Rope pulls with tension T towards itself at each end • Pulleys change direction of rope without changing tension • Force on pulley axle is (vector) sum of tension of rope on each side • Circular motion • Fc = -mv2/r ř

  3. (Non-) Uniform Circular Motion • What about swinging the ball about vertically? • The speed is no longer constant! • What is tension? • Use radial coordinate system

  4. Unit vectors in Polar Coordinates • For uniform circular motion we have • What’s v? • Unit vectors aren’t constant!

  5. (Non-) Uniform Circular Motion • What’s the slowest the ball can go at top, and stay on circle?

  6. Non-Inertial Reference Frames • When reference frame is non-inertial (accelerating, rotating) apparent forces arise • “Fictitious forces” • Will counteract acceleration of frame • Consider an accelerating train car with object hanging from ceiling • Observer on ground says horizontal component of T accelerates the object • Observer on train says horizontal component of T ballences a horizontal force pulling object back

  7. Non-Inertial Reference Frames • For rotating reference frames, fictitious force is called “centrifugal force” • It’s the “force” which “pulls” an object away from the center • Actually, friction provides centripetal acceleration to keep object moving in a circle

  8. Non-Inertial Reference Frames • In rotating frame, an object “feels” the “Coriolis force” perpendicular to its velocity

  9. Resistive Forces • Motion (on Earth) usually gives rise to resistive forces • No (relative) motion, no resistive force • Gives rise to misconception that force is necessary for movement • Friction: independent of velocity (in kinetic regime) • Drag: in air or water depends on velocity • Complicated function of velocity, but two simple cases are studied: linear and quadratic

  10. Resistive Forces • Linear Case F = -bv • Terminal Velocity • Solution for v0 = 0

  11. Resistive Forces • Quadratic Case F = ½DrAv2 • r density • A cross-sectional area • D drag coefficient (0.5 for sphere) • Terminal Velocity

  12. Numerical Modeling • Specifying the forces should determine position, velocity of particle at all times • But it’s not always easy to solve analytically • Can try to find answer numerically • Euler’s Method • Break time into (small) discrete steps: Dt • Specify starting conditions (position, velocity) • Find acceleration at current position (and velocity) from ai = F(xi,vi,ti) • Find next position from xi+1 = xi+viDt • Find next velocity from vi+1 = vi+aiDt • Repeat from step 3.

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