1 / 32

PHY1012F VECTORS

PHY1012F VECTORS. Gregor Leigh gregor.leigh@uct.ac.za. VECTORS. Resolve vectors into components and reassemble components into a single vector with magnitude and direction. Make use of unit vectors for specifying direction.

moriah
Download Presentation

PHY1012F VECTORS

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. PHY1012FVECTORS Gregor Leighgregor.leigh@uct.ac.za

  2. VECTORS • Resolve vectors into components and reassemble components into a single vector with magnitude and direction. • Make use of unit vectors for specifying direction. • Manipulate vectors (add, subtract, multiply by a scalar) both graphically (geometrically) and algebraically. Learning outcomes:At the end of this chapter you should be able to…

  3. VECTORS • Using only positive and negative signs to denote the direction of vector quantities is possible only when working in a single dimension (i.e. rectilinearly). • In order to deal with direction when describing motion in 2-d (and later, 3-d) we manipulate vectors using either graphical (geometrical) techniques, or the algebraic addition of vector components.

  4. SCALARS and VECTORS • Scalar – A physical quantity with magnitude (size) but no associated direction.E.g. temperature, energy, mass. Vector – A physical quantity which has both magnitude AND direction.E.g. displacement, velocity, force.

  5. Algebraically, we shall distinguish a vector from a scalar by using an arrow over the letter, . VECTOR REPRESENTATION and NOTATION • Graphically, a vector is represented by a ray.The length of the ray represents the magnitude, while the arrow indicates the direction. The important information is in the direction and length of the ray – we can shift it around if we do not change these. Note: r is a scalar quantity representing the magnitude of vector , and can never be negative.

  6. GRAPHICAL VECTOR ADDITION • A helicopter flies 15 km on a bearing of 10°, then 20 km on a bearing of 250°. Determine its net displacement. N 20 km 60° 10° 15 km   = 74°

  7. MULTIPLYING A VECTOR BY A SCALAR • Multiplying a vector by a positive scalar gives another vector with a different magnitude but the same direction: Notes: • B = cA. (c is the factor by which the magnitude ofis changed.) • lies in the same direction as . • (Distributive law). • If c is zero, the product is the directionless zero vector, or null vector.

  8. y • x • z VECTOR COMPONENTS • Manipulating vectors geometrically is tedious. • Using a (rectangular) coordinate system, we can use components to manipulate vectors algebraically. • We shall use Cartesian coordinates, a right-handed system of axes: (The (entire) system can be rotated – any which way – to suit the situation.)

  9. VECTOR COMPONENTS • Adding two vectors (graphically joining them head-to-tail) produces a resultant (drawn from the tail of the first to the head of the last)…

  10. VECTOR COMPONENTS • Adding two vectors (graphically joining them head-to-tail) produces a resultant (drawn from the tail of the first to the head of the last)… “Running the movie backwards” resolves a single vector into two (or more!) components.

  11. VECTOR COMPONENTS • Adding two vectors (graphically joining them head-to-tail) produces a resultant (drawn from the tail of the first to the head of the last)… “Running the movie backwards” resolves a single vector into two (or more!) components.

  12. VECTOR COMPONENTS • Adding two vectors (graphically joining them head-to-tail) produces a resultant (drawn from the tail of the first to the head of the last)… “Running the movie backwards” resolves a single vector into two (or more!) components. ??! Even if the number of components is restricted, there is still an infinite number of pairs into which a particular vector may be decomposed. Unless…

  13. VECTOR COMPONENTS • …by introducing axes, we specify the directions of the components. • y is now constrained to resolve into and , at right angles to each other. Note that, provided that we adhere to the right-handed Cartesian convention, the axes may be orientated in any way which suits a given situation. • x

  14. y • x VECTOR COMPONENTS • Resolution can also be seen as a projection of onto each of the axes to produce vector components and . Ax, the scalar component of (or, as before, simply its component) along the x-axis … • has the same magnitude as . • is positive if it points right; negative if it points left. • remains unchanged by a translation of the axes (but is changed by a rotation).

  15. y (m) 8 6 4 2 • x (m) 0 0 2 4 6 8 VECTOR COMPONENTS • The components of are… Ax = +6 m Ay = +3 m

  16. VECTOR COMPONENTS • y (m) • The components of are… 8 6 Ax = +6 m Ay = +3 m 4 2 • x (m) -8 -6 -4 -2 0

  17. VECTOR COMPONENTS • y (m) • The components of are… -8 -6 -4 -2 • x (m) -2 Ax = +6 m Ay = +3 m -4 -6 -8

  18. VECTOR COMPONENTS • y (m) • The components of are… 4 2 Ax = –2 m Ay = +4 m • x (m) -4 -2 2 -2

  19. VECTOR COMPONENTS • y (m) • The components of are… 4 2 Ax = –6 m Ay = –5 m • x (m) -8 -4 4 -2

  20. VECTOR COMPONENTS The components of are… PHY1012F • y (m) 4 • x (m) 2 Ax = –6 m Ay = –5 m –8 m –3 m 4 -4 -8 -4 20

  21. PHY1010W VECTOR COMPONENTS • y (m) • The components of are… • x (m) Ax = +Acos m Ay = –Asin m • 

  22. PHY1010W VECTOR COMPONENTS • y (m) • The components of are… • x (m) Ax = –Asin m Ay = –Acos m •  Note that we can (re)combine components into a single vector, i.e. (re)write it in polar notation, by calculating its magnitude and direction using Pythagoras and trigonometry: (On this slide!)

  23. y 1 • x 1 UNIT VECTORS • Components are most useful when used with unit vector notation. • A unit vector is a vector with a magnitude of exactly 1 pointing in a particular direction: • A unit vector is pure direction – it has no units! • Vector can now be resolved and written as:

  24. UNIT VECTORS • y (m/s) Given a 12 m/s velocity vector which makes an angle of 60° with the negative x-axis, write the vector in terms of components and unit vectors. vy = +vsin60° v = 12 m/s • 60° vx = –(12 m/s)cos60° vx= –6.00 m/s • x (m/s) vx = –vcos60° vy = +(12 m/s)sin60° vy= +10.4 m/s Hence:

  25. ALGEBRAIC ADDITION OF VECTORS • Suppose and Thus Dx= Ax + Bx + Cx and Dy= Ay + By + Cy In other words, we can add vectors by adding their components, axis by axis, to determine a single resultant component in each direction. These resultants can then be combined, or simply presented in unit vector notation.

  26. y • x ALGEBRAIC ADDITION OF VECTORS • The process of vector addition by the addition of components can visualised as follows: By Dy = Ay + By Ay = + Ax Bx Dx = Ax + Bx

  27. y • x ALGEBRAIC ADDITION OF VECTORS • While it is often quite acceptable to present as Dy = Ay + By its polar form is easily reconstituted from Dx and Dy using and •  Dx = Ax + Bx

  28. GRAPHICAL VECTOR ADDITION • A helicopter flies 15 km on a bearing of 10°, then 20 km on a bearing of 250°. Determine its net displacement. N 20 km 60° 10° 15 km   = 74°

  29. ALGEBRAIC ADDITION OF VECTORS • A helicopter flies 15 km on a bearing of 10°, then 20 km on a bearing of 250°. Determine its net displacement. • y N 20 km +15 sin10° +2.60 +15 cos10° +14.77 10° –18.79 –20 cos20° –20 sin20° –6.84 15 km –16.2 +7.93  • x 20°

  30. N • y • x ALGEBRAIC ADDITION OF VECTORS • A spelunker is surveying a cave. He follows a passage 100 m straight east, then 50 m in a direction 30° west of north, then 150 m at 45° west of south. After a fourth unmeasured displacement he finds himself back where he started. Determine the magnitude and direction of his fourth displacement. 30° 50 m 100 m 150 m 45°

  31. ALGEBRAIC ADDITION OF VECTORS • y N Rx = 100 – 25 –106 + Dx = 0  Dx = 31 30° Ry = 0 +43.3 –106 + Dy = 0  Dy = 62.7 50 m • x 100 m 150 m 45° 100 0 –25 43.3 –106 –106 63.7 69.9 ? ? ? 31 62.7 ?

  32. VECTORS • Resolve vectors into components and reassemble components into a single vector with magnitude and direction. • Make use of unit vectors for specifying direction. • Manipulate vectors (add, subtract, multiply by a scalar) both graphically (geometrically) and algebraically. Learning outcomes:At the end of this chapter you should be able to…

More Related