1 / 24

Chapter 3

Chapter 3. Vectors. Frame of Reference. In order to describe motion, you have to pick a “frame of reference” Frame of reference is always decided by the observer In physics we often use a coordinate axis to denote our frame of reference. Frame of Reference. Y. -x. x.

matia
Download Presentation

Chapter 3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 3 Vectors Herriman High AP Physics C

  2. Frame of Reference • In order to describe motion, you have to pick a “frame of reference” • Frame of reference is always decided by the observer • In physics we often use a coordinate axis to denote our frame of reference. Herriman High AP Physics C

  3. Frame of Reference Y -x x This is a familiar coordinate axis, but in physics it has a slightly different meaning. -Y Herriman High AP Physics C

  4. Frame of Reference • In this frame of reference, up is positive, right is positive • Down in negative, as is left • Hence in physics a negative has no value, it merely denotes direction • This is necessary because direction has a special distinction in physics; it separates vectorquantities fromscalarquantities Herriman High AP Physics C

  5. Vector vs. Scalar • A Scalar Quantity is one that has only magnitude – distance is a scalar • Example: If you travel 500 miles that is a distance • A Vector Quantity is one that has magnitude and direction – displacement is a vector • Example: If you travel 500 miles North this is displacement Herriman High AP Physics C

  6. Vector vs. Scalar • If you divide distance by time you get average speed • Example: S = D/t = 500 miles/2 hours =250 mph • If you divide displacement by time you get average velocity • Example: Vavg = x/t = 500 miles North/2 hours = 250 mph North Herriman High AP Physics C

  7. Vector Addition • Two Methods • Graphical Method • Requires a ruler and a protractor • Process • Convert each vector into a line that fits a scale of your choosing • Draw Vectors head to tail, measuring the angle exactly • Draw a resultant • Measure the resultant and the resulting angle • Convert the measurement back into a vector answer Herriman High AP Physics C

  8. Graphical Addition • A man drives 125 km west and then turns 45º North of West and travels an additional 100 km. What is his displacement? • Step one: Set Scale • 1 cm = 25 Km 100 km = 4 cm 45° Herriman High AP Physics C 125 km = 5 cm

  9. Graphical Addition Step Two: Move Vectors Head to Tail Step Three: Draw Resultant from Tail of First Vector to the head of the last. 100 km = 4 cm 45° 125 km = 5 cm Herriman High AP Physics C

  10. Graphical Addition Step Four: Measure the Measure the Resultant and the resulting angle. Step Five: Using the Scale, convert the measurement into an answer. 100 km = 4 cm 8.3 cm 8.3 cm * 25 Km/cm = 208 Km 45° 125 km = 5 cm 19.9° Final Answer: His displacement is 208 Km @ 19.9° North of West Herriman High AP Physics C

  11. Mathematical Addition • Mathematical Addition of Vectors Requires a basic knowledge of Geometry – You must know: • Sin θ = Opposite/Hypotenuse • Cos θ = Adjacent/Hypotenuse • Tan θ = Sin θ/ Cos θ =Opposite/Adjacent • ArcSin, ArcCos, ArcTan Herriman High AP Physics C

  12. Mathematical Addition • Mathematical addition requires that you be able to • Draw a rough sketch of the original vectors • Resolve each vector into its Components • Reduce the component vectors into a system of two vectors • Draw a parallelogram • Draw the Resultant • Use Triangle geometry to find the magnitude and direction of the resultant Herriman High AP Physics C

  13. Mathematical Addition Step One: Draw Original Vectors 45º 125 Km West 100 Km 45° North of West Herriman High AP Physics C

  14. Mathematical Addition Step Two: Resolve Vectors into components Cos θ = W/100 km W = 100 Cos 45° =70.7 Km Sin θ = N/100 km N = 100 Sin 45° =70.7 Km 125 Km West 100 Km 45° North of West Herriman High AP Physics C

  15. Mathematical Addition Step Three: Reduce to a system of two vectors 195.7 Km West 70.7 Km North 70.7 Km North 70.7 Km West = + 125 Km West Herriman High AP Physics C

  16. B A Mathematical Addition • Step 4: Draw Parallelogram • Step 5: Draw Resultant • Step 6: Use Right Triangle Geometry Pythagorean Theorem: C2 = A2 + B2 C = SQRT(A2 + B2) = SQRT{(197.5)2+(70.7)2} = 209.8 km Tan θ = 70.7/197.5 = 0.3579 Θ= ArcTan 0.3579 = 19.7° North or West Herriman High AP Physics C

  17. Graphical vs. Mathematical Methods • Graphical Method • Less Complicated Math • Good Approximation • Not as exact as you would like • Mathematical Method • No need for ruler or protractor • Less time dedicated to sketch • More Accurate result • Mathematics is more difficult Herriman High AP Physics C

  18. Component Vectors Unit Vectors • The unit vector in the x direction is denoted by i • The unit vector in the y direction is denoted by j • The unit vector in the z direction is denoted by k Rule: i2 = j2 = k2 = 1 Herriman High AP Physics C

  19. B A Component Vectors Unit Vectors • In terms of component unit vectors: • Vector A =Cx= 5i • Vector B = Cy = 3j • Hence Vector C = 5i + 3j 5.83 units 3 units C 5 units Herriman High AP Physics C

  20. Adding Vectors Component Method • Vector A = 2i + 3j - 2k • Vector B = 5i- 6j + 8k • A + B = 7i – 3j + 6k Magnitude of the Vector |A + B| = SQRT{(7i)2+(-3j)2+(6k)2} = 9.7 Rules of Vector Addition • A + B = B + A • A – B = A + (-B) Herriman High AP Physics C

  21. Multiplication of Vectors • Three types of Vector Multiplication • Multiply a vector by a scalar • Vector A = 3i + 2j + 5k • Vector 4A = 12i + 8j + 20k Herriman High AP Physics C

  22. Multiplication of Vectors • Multiply 2 Vectors to get a scalar (dot product) • a•b = |a||b| cos θ = axbx+ ayby + azbz Rule: i•i=j•j = k•k = 1 i•j = i•k = j•k = 0 • Vector A = 2i + 3j - 2k • Vector B = 5i- 6j + 8k • a•b =(2i*5i) + (3j * -6j) + (-2k * 8k) = 10 – 18 – 16 = -24 Herriman High AP Physics C

  23. Multiplication of Vectors • Multiply 2 Vectors to get a Vector (cross product) • A x b = |a||b| sin θ = axby+ axbz + aybx + aybz + azbx+ azby Rule: i x i=j x j = k x k = 0 i x j = k ; j x k = i ; k x i = j j x i = -k ; k x j = -i ; i x k = -j Herriman High AP Physics C

  24. Vector A = 2i + 3j - 2k Vector B = 5i- 6j + 8k A x B = (2i x -6j) + (2i x 8k) + (3j x 5i) + (3j x 8k) + (-2k x 5i) + (-2k x -6j) -12k + - 16j + -15k + 24i + -10j + -12i 12i - 26j – 27k Multiplication of Vectors Herriman High AP Physics C

More Related