straight line.u4.outcome1

7/30/2019 Straight Line.U4.Outcome1

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Example(1):


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Example(2):


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Example(3):

Example(4):

Example(5):


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Example(6):


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Example(7):


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Example(8):

Example(9):


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Example(10):


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Example(11):


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Straight Line

Directly Proportional Relationships

In Engineering and Science, the Relationship between two quantities is often Directly Proportional and

when one is plotted against the other, a straight line graph through the origin is produced

Example(1):

Mechanical Spring

Change in length = xL

Applied Force=F

Then direct Proportionality implies:

xF

mxF (1)

If, for example:

NewtonF .1 , when, mmx 2 (for a particular spring) , then, substituting into Eq.1 gives:

2.1 m mmNm /502

1

The spring equation becomes:

xF 50

linestraighteachfortconsaisitandSlopeorGradientm ....tan.......


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Figure for Example(1)

Example (2):

Consider the graph that relates the Fahrenheit and Celsius temperature scales.

This graph can be used to convert from one temperature scale to the other.

Given that :

Fy0.212 for Cx 0.100 this is one point on the straight line relating the two scales )212,100( .

Fy 0.32 for Cx 0.0 this is a second point on the straight line relating the two scales )32,0( .

Knowing that the straight line is completely and uniquely determined by two points on it, one gets the

straight line relating the two scales by joining the above two points.


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Figure for Example(2)

Let FTFahrenheitetemperatur . ,

CTCelsiusetemperatur .

The slope of the straight line =

))((

))((

1

1

CC

FF

TT

TTm

))()((

))()((

12

12

CC

FF

TT

TTconstant

))()((

))()((

12

12

CC

FF

TT

TT

m 5

9

10

18

100

180

)0100(

)32212(

Also ,

5

9

)0(

)32(

C

F

T

Tm

5

9)32(

C

F

T

T CF TT

5

932 32

5

9 CF TT (1)

Equation (1) is an equation of a straight line in the form:

Cmxy

where;

5

9)..(. mSlopeorGradientThe

32. CInterceptyThe


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Note:

The relationship between CF TandT .. is not direct proportionality, rather , the relationship between

alityproportiondirectisTandT CF ....).32( ; i.e.

..)32( CF TT

.5

9)32( CF TT

Let

)32(* FF TT

Then

CF TT )5

9(* which is a direct proportionality relationship.

Therefore, CF TandT ..*

are directly proportional.

Example (3):

The equation of the straight line in the form:

Cmxy

will be:

CmP

Where,


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)(

)(

12

12

PPm

)0300(

)100210( 300

11030

11= 3670

100. CInterceptyThe because, 0...100 whenP

Therefore, the equation of the straight line becomes:

1003670 P

Now find for 0P which is the value of of the point of intersection of the line with the - axis.

10036700 C052723670

100

Which is the same as the value ofat the intersection of the extrapolated line with the axis

Example (4):

The graph below shows a linear relationship between physical quantities ( two variables).

Deduce the law that governs their relationship.


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The equation of the straight line is of the form:

Cmxy

This graph can be used to find Candm ..

From the graph, one finds that :

30y for 517 x this is one point on the line relating the two variables .

Cerceptyy int15 for 0x this is a second point on the straight line relating the two

variables.


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)(

)(

12

12

xx

yym

)0517(

))15(30(

517

4535

90= 5712

Therefore, the equation of the straight line becomes:

15)3590( xy (exact)

15571.2 xy (rounded off)

Inverse Proportionality:

The )/1( x button on calculators is called the (inverse button).

If one enters ( 5) and press )/1( x button one gets (0.2) because it evaluates51

.

Thus inverse means (1 divided by). So ,x

xinverse1

)..(

Examples of Inverse Proportionality:

1. Density )( and Volume )(V for a given (constant) Mass )(m :

V

m 1 ;

When one plots

Vagainst

1.. one gets a straight line passing through the origin.

2. Pressure )(P and Volume )(V for a given (constant) Temperature ( T):

V

CP1

;

When one plots

VagainstP

1.. one gets a straight line passing through the origin.


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Example (5):

1. Determine the law relating P and V from the graph below.2. Determine the pressure when the volume, 350 mV .

The law is :

VCP

1

This happens when V physically this means that P goes to zero as V assumes very large

values.

01

...01

1

VwhenP

3

2

2

2 31

....1000

m

VwhenmNP


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12

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VV

PPmC

)03(

)01000(

3

1000

Nm3333 The relating law becomes:

VP

13333 Or, equivalently 3333PV

Example (6):


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Example (7):

Solution:

straight line.u4.outcome1

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