phy1012f vectors

NEWTON’S LAWS VECTORS

PHY1012F

VECTORS

Gregor [email protected]

NEWTON’S LAWS VECTORSPHY1012F

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VECTORSLearning outcomes:

At the end of this chapter you should be able to…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.


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VECTORSUsing 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.


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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.


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VECTOR REPRESENTATION and NOTATIONGraphically, a vector is represented by a ray.

The length of the ray represents the magnitude, while the arrow indicates the direction.

r

Algebraically, we shall distinguish a vector from a scalar by using an arrow over the letter, .r

Note: r is a scalar quantity representing the magnitude of vector , and can never be negative.

r

The important information is in the direction and length of the ray – we can shift it around if we do not change these.


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GRAPHICAL VECTOR ADDITIONA helicopter flies 15 km on a bearing of 10°, then 20 km on a bearing of 250°. Determine its net displacement.

N

10°

15 km

20 km

= 74°

60°

R

2 2 215 20 2 15 20 cos60R

18 kmR

sin sin6020 18

18 km, 296R


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MULTIPLYING A VECTOR BY A SCALARMultiplying a vector by a positive scalar gives another vector with a different magnitude but the same direction:B cA

Notes: B = cA. (c is the factor by which the magnitude of is changed.) lies in the same direction as . (Distributive law).If c is zero, the product is the directionless zero vector, or null vector.

A

B

A

c A Z cA cZ

A

1B c

A

1B c


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VECTOR COMPONENTSManipulating 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: y

x

z

(The (entire) system can be rotated – any which way – to suit the situation.)


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VECTOR COMPONENTSAdding two vectors (graphically joining them head-to-tail) produces a resultant (drawn from the tail of the first to the head of the last)…

1 2A A A

1A

2A


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A

“Running the movie backwards” resolves a single vector into two (or more!) components.

1A

2A


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A

“Running the movie backwards” resolves a single vector into two (or more!) components.

2A

3A

1A


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A

“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…

1A

2A

1A

2A

1A

2A

??!


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VECTOR COMPONENTS…by introducing axes, we specify the directions of the components.

y

xxA

yA

is now constrained to resolve into and , at right angles to each other.

A

xA

yA

A

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.

xA

yA


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VECTOR COMPONENTSResolution can also be seen as a projection of onto each of the axes to produce vector components and .

y

x

A

xA

yA

Ax, the scalar component of (or, as before, simply its component) along the x-axis …

A

has the same magnitude as .xA

remains unchanged by a translation of the axes (but is changed by a rotation).

is positive if it points right; negative if it points left.

A

xA

yA


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VECTOR COMPONENTSThe components of are…

A y (m)

2

4

0

6

8

2 4 60 8x (m)

A

xA

yAAx = +6 m

Ay = +3 m


16

0


A

y (m)

2

4

6

8

-6 -4 -2-8x (m)

A

yAAx = +6 m

Ay = +3 mxA


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A y (m)

-6

-4

-2

-6 -4 -2-8x (m)

A

yAAx = +6 m

Ay = +3 m

xA

-8


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A y (m)

-2

2

4

-4 -2 2x (m)

A

xA yA

Ax = –2 m

Ay = +4 m


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A

y (m)

-2

2

4

-8 -4 4x (m)

A

xA

yA

Ax = –6 m

Ay = –5 m


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A

y (m)

2

4

-8-4

4

x (m)

A

xA

yAAx = –6 m

Ay = –5 m-4

–8 m

–3 m

NEWTON’S LAWS VECTORSPHY1012FPHY1010W

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A

y (m)

x (m)

A

xA

Ax = +A cos m

Ay = – A sin m

yA

NEWTON’S LAWS VECTORSPHY1012FPHY1010W

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A y (m)

x (m)

A

xA

yAAx = –A sin m

Ay = – A cos 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:

2 2x yA A A 1tan x

y

AA

(On this slide!)


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UNIT VECTORSComponents 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:

y

x1

1

i

j

A unit vector is pure direction – it has no units!

i 1, + -directionx j 1, + -directiony

Vector can now be resolved and written as:A

ˆ ˆi jx y x yA A A A A

k 1, + -directionz


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UNIT VECTORSy (m/s)

x (m/s)

v

60°

vx = –(12 m/s) cos60° vx = –6.00 m/s

vx = –v cos60°

v = 12 m/s

vy = +v sin60°

vy = +(12 m/s) sin60° vy = +10.4 m/s

ˆ ˆ6.00i 10.4j m/sv Hence:

i

j

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.


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ALGEBRAIC ADDITION OF VECTORS Suppose D A B C

ˆ ˆ ˆ ˆ ˆ î j i j i jx y x y x yA B C A A B B C C

ˆ î jx x x y y yA B C A B C A B C

ˆ î jx yD D D

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.

and


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y

x+

ALGEBRAIC ADDITION OF VECTORSThe process of vector addition by the addition of components can visualised as follows:

A

D B

Ax Bx

By

Ay

Dy =

Ay +

By

Dx = Ax + Bx

A

= D

B


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ALGEBRAIC ADDITION OF VECTORSWhile it is often quite acceptable to present as

A

D

y

x

B

Dy =

Ay +

By

Dx = Ax + Bx

D

ˆ ˆi jx yD D D

its polar form is easily reconstituted from Dx and Dy

using

and

2 2x yD D D

1tan y

x

DD


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GRAPHICAL VECTOR ADDITIONA helicopter flies 15 km on a bearing of 10°, then 20 km on a bearing of 250°. Determine its net displacement.

N

10°

15 km

20 km

= 74°

60°

R

2 2 215 20 2 15 20 cos60R

18 kmR

sin sin6020 18

18 km, 296R


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ALGEBRAIC ADDITION OF VECTORSA helicopter flies 15 km on a bearing of 10°, then 20 km on a bearing of 250°. Determine its net displacement.N

10°

15 kmR

ˆ ˆ16.2i 7.93j km R 20°

x-comp’nt (km)

y-comp’nt (km)

A

BA

y

x R

2 2 18 kmx yR R R 1tan 26 (ie bearing 296 )y

x

RR

B 20 km +15 sin10° +15 cos10°

–20 cos20° –20 sin20°

–16.2 +7.93

+2.60 +14.77

–18.79 –6.84


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Ny

x

ALGEBRAIC ADDITION OF VECTORSA spelunker is surveying a cave. He follows a passage 100 m straight east, then 50 m in a direction 30° west of north, then

30°

45°

A

B

50 m

100 m150 m

C

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.


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Rx = 100 – 25 –106 + Dx = 0

ALGEBRAIC ADDITION OF VECTORSN

30°

45°

y

x50 m

100 m150 m

A

B

C

Vector Magntd (m)

Angle x-comp’nt (m) y-comp’nt (m)

1000°

50 120°

150 225°

Rx = 0 Ry = 0

A

B

C

D

R

Ry = 0 +43.3 –106 + Dy = 0

2 231 62.7 69.9 mD

1 62.7tan 63.731

Dx = 31

Dy = 62.7

100 0–25 43.3

–106 –10631 62.7? ?69.9 63.7? ?


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VECTORSLearning outcomes:

At the end of this chapter you should be able to…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.

phy1012f vectors

Documents

magnitude of vector

single vector

direction of vector

null vector

vectors scalar

associated direction

use of unit vectors

scalar quantity