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Measurement and Vectors in Physics

This lecture covers units, significant figures, density measurements, estimation, vectors, and vector addition in physics.

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Measurement and Vectors in Physics

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  1. Phys 201, Spring 2011Chapt. 1, Lect. 2 • Reminders: • The grace period of WebAssign access is two weeks. • 2. The lab manual can be viewed online, but you are • recommended to buy a hard copy from the bookstore. • For honors credit, attend Friday’s 207 lecture, • TOMORROW 12:05pm, (or write an essay later). Chapter 1: Measurement and vectors Phys 201, Spring 2011

  2. Review from last time:  Units: Stick with SI, do proper conversion. (in each equation, all terms must match!) • Dimensionality: any quantity in terms of L, T, M (the dimension for basic units). • Significant figures (digits) and errors • Estimate (rounding off to integer) and order of magnitude (to 10’s power) Phys 201, Spring 2011

  3. The density of seawater was measured to be 1.07 g/cm3. This density in SI units is • 1.07 kg/m3 • (1/1.07) x 103 kg/m3 • 1.07 x 103 kg • 1.07 x 10–3 kg • 1.07 x 103 kg/m3 Phys 201, Spring 2011

  4. The density of seawater was measured to be 1.07 g/cm3. This density in SI units is • 1.07 kg/m3 • (1/1.07) x 103 kg/m3 • 1.07 x 103 kg • 1.07 x 10–3 kg • 1.07 x 103 kg/m3 An order of magnitude estimate: 1 tonne /m3 Phys 201, Spring 2011

  5. If K has dimensions ML2/T2, then k in the equation K = kmv2 must • have the dimensions ML/T2. • have the dimension M. • have the dimensions L/T2. • have the dimensions L2/T2. • be dimensionless. Phys 201, Spring 2011

  6. If K has dimensions ML2/T2, then k in the equation K = kmv2 must • have the dimensions ML/T2. • have the dimension M. • have the dimensions L/T2. • have the dimensions L2/T2. • be dimensionless. Phys 201, Spring 2011

  7. Today: Vectors • In one dimension, we can specify distance with a real number, including + or – sign (or forward-backward). • In two or three dimensions, we need more than one number to specify how points in space are separated – need magnitude and direction. Madison, WI and Kalamazoo, MI are each about 150 miles from Chicago. Phys 201, Spring 2011

  8. Denoting vectors Scalars, vectors: Scalars are those quantities with magnitude, but no direction: Mass, volume, time, temperature (could have +- sign) … Vectors are those with both magnitude AND direction: Displacement, velocity, forces … • Two of the ways to denote vectors: • Boldface notation: A • “Arrow” notation: Phys 201, Spring 2011

  9. Example: Displacement (position change) Phys 201, Spring 2011

  10. Adding displacement vectors Phys 201, Spring 2011

  11. “Head-to-tail” method for adding vectors Phys 201, Spring 2011

  12. Vector addition is commutative Vectors “movable”: the absolute position is less a concern. Phys 201, Spring 2011

  13. Adding three vectors:vector addition is associative. Phys 201, Spring 2011

  14. A vector’s inverse has the same magnitude and opposite direction. Phys 201, Spring 2011

  15. Subtracting vectors Phys 201, Spring 2011

  16. Example 1-8. What is your displacement if you walk 3.00 km due east and 4.00 km due north? Pythagorean theorem Phys 201, Spring 2011

  17. Components of a vector along x and y Phys 201, Spring 2011

  18. Components of a vector:along an arbitrary direction Phys 201, Spring 2011

  19. Magnitude and direction of a vector Phys 201, Spring 2011

  20. Adding vectors using components Cx = Ax + Bx Cy = Ay + By Phys 201, Spring 2011

  21. Unit vectors A unit vector is a dimensionless vector with magnitude exactly equal to one. The unit vector along x is denoted The unit vector along y is denoted The unit vector along z is denoted aaa Phys 201, Spring 2011

  22. Which of the following vector equations correctly describes the relationship among the vectors shown in the figure? Phys 201, Spring 2011

  23. Which of the following vector equations correctly describes the relationship among the vectors shown in the figure? Phys 201, Spring 2011

  24. Can a vector have a component bigger than its magnitude? Yes No Phys 201, Spring 2011

  25. Can a vector have a component bigger than its magnitude? • Yes • No The square of a magnitude of a vector R is given in terms of its components by R2 = Rx2 + Ry2 . Since the square is always positive, no component can be larger than the magnitude of the vector. Thus, a triangle relation: | A | + | B | > | A+B | Phys 201, Spring 2011

  26. Properties of vectors: summary Phys 201, Spring 2011

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