Chap 1 Cutnell

8/13/2019 Chap 1 Cutnell

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In t roduc t ion and

Mathematical Concep ts

Chapter 1


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Tr igonometry


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Tr igonometry

h

hosin

h

ha

cos

a

o

h

htan


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Tr igonometry

m2.6750tan oh

m0.80m2.6750tan oh

a

o

h

h

tan


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Tr igonometry

h

ho1sin

h

ha1cos

a

o

h

h1tan


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Tr igonometry

a

o

h

h1tan 13.9m0.14

m25.2tan 1


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Tr igonometry

222

ao hhh Pythagorean theorem:


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Scalars and Vectors

A scalar quantity is one that can be described

by a single number:

temperature, speed, mass

A vector quantity deals inherently with both

magnitude and direction:

velocity, force, displacement


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Scalars and Vectors

By convention, the length of a vectorarrow is proportional to the magnitude

of the vector.

8 lb4 lb

Arrows are used to represent vectors. The

direction of the arrow gives the direction ofthe vector.


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Vector Addi t ion and Subtract ion

Often it is necessary to add one vector to another.


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Vector A ddi t ion and Subtract ion

5 m 3 m

8 m


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2.00 m

6.00 m


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Vector A ddi t ion and Subtract ion

2.00 m

6.00 m

222 m00.6m00.2 R

R

m32.6m00.6m00.2 22 R


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2.00 m

6.00 m

6.32 m

00.600.2tan

4.1800.600.2tan 1


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When a vector is multiplied

by -1, the magnitude of the

vector remains the same, but

the direction of the vector isreversed.


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A

B

BA

A

B

BA


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The Components of a Vecto r

.ofcomponentvectortheand

componentvectorthecalledareand

r

yx

y

x


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.AAA

AA

A

yx

thatsoyvectorialltogetheraddand

axes,andthetoparallelarethatandvectors

larperpendicutwoareofcomponentsvectorThe

yxyx


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It is often easier to work with the scalar components

rather than the vector components.

.of

componentsscalartheareand

A

yx AA

1.magnitudewithrsunit vectoareand yx

yxA yx AA

C f


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The Components of a Vector

Example

A displacement vector has a magnitude of 175 m and points atan angle of 50.0 degrees relative to thexaxis. Find thexand y

components of this vector.

rysin

m1340.50sinm175sin ry

rxcos

m1120.50cosm175cos

rx

yxr m134m112

Addi t i f V t b M f C t


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Addi t ion of Vectors by Means of Components

BAC

yxA yx AA

yxB yx BB

Addi t i f V t b M f C t


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Addi t ion of Vectors by Means of Components

yxyxyxC

yyxx

yxyx

BABA

BBAA

xxx BAC yyy BAC


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Problems.

Two workers are trying to move a heavy crate. One

pushes on the crate with a force A , which has a

magnitude of 428 newtons (N) and is directed due

west. The other pushes with a force B , which has a

magnitude of 192 N and is directed due north. What

are (a)the magnitude and (b)direction of the resultant

force A+ B applied to the crate? Suppose that the

second worker applies a force B instead of B. What

then are (c)the magnitude and (d)direction of the

resultant force A- B applied to the crate? In both cases

express the direction as a positive angle relative to duewest.


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The displacement vector has scalar

components ofAx= 80.0 m andAy= 60.0m. The displacement vector has a scalar

component of Bx= 60.0 m and a

magnitude of B= 75.0 m. The

displacement vector has a magnitude of C

= 100.0 m and is directed at an angle of

36.9 above the +xaxis. Determine which

two of these vectors are equal.


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Three forces are applied to an object, as shown in the figure. Force

F1has a magnitude of 20.6 newtons (N) and is directed 30.0 to the

left of the +yaxis. Force F2has a magnitude of 13.0 N and pointsalong the +xaxis. A third force F3is applied such that the vector

sum of the three forces is 0 N. What are (a)the magnitude and (b)

direction of this third force ? Specify the direction as a positive

angle relative to the negativexaxis, as shown in the figure.

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Chap 1 Cutnell