PHYSICS

MATH SKILLS

BOOKLET

2012

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PRACTICE, PRACTICE, PRACTICE

You require certain maths skills to ensure your success in Stage 2 Physics. It is easy to forget basic skills;

this will prevent you from successfully solving physics problems. The good news is that practice assists the

retention of these skills, so practice, practice, practice!

It is likely that you may have forgotten some of these essential skills. It may be that you never learnt that

skill in the first place. When in doubt head straight to the YouTube or the Khan Academy website (do not

get distracted by facebook!) to look for a tutorial.

If you find the problems so hard you can’t do them or get them wrong - it is not a bad thing – it is only bad

if you fail to seek help. Getting problems wrong is a good thing – it helps you identify areas you need to

work on – mistakes are your friend – come and see me and make me feel useful!

It is possible that you may need refresher through the year – you can keep testing yourself by coming back

to this booklet. Questions are rated for difficulty - from easy to difficult to help you gauge your

level of understanding. You will be given an exercise book to do the answers in – keep the question booklet

with your answer booklet. At the end of each set self-evaluate.

TABLE OF CONTENTS SET A - Rearranging Equations ........................................................................................................................... 2

SET B - Significant figures ................................................................................................................................... 3

SET C - Rounding ................................................................................................................................................ 3

SET D - Scientific notation .................................................................................................................................. 4

SET E - Using Pythagoras Theorem .................................................................................................................... 5

SET F - Trignonometry ........................................................................................................................................ 6

SET G – Representing Vectors ............................................................................................................................ 7

SET H – Vector Addition and Subtraction .......................................................................................................... 7

SET I - Unit Conversions SI base units ................................................................................................................ 8

SET J - Slope of a line. ......................................................................................................................................... 9

SET K - Equation of a straight line. ................................................................................................................... 11

Set L - Interpreting graphs. (Proportional relationships) ................................................................................. 13

Self-evaluation

The problems too hard and I couldn’t do them –

I need HELP!

The problems were more hard than easy

The problems were more easy than hard

No problemo – too easy!

Self-evaluation

The problems too hard and I couldn’t do them – I need HELP!

The problems were more hard than easy

The problems were more easy than hard

No problemo – too

easy!

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SET A - REARRANGING EQUATIONS

RULE #1: You can add, subtract, multiply and divide by anything in an equation, as long as you do the same

thing to both sides of the equals sign. The equals sign acts like the fulcrum of a balance: if you add 5 of

something to one side of the balance, you have to add the same amount to the other side to keep the

balance steady.

RULE #2 When rearranging, make sure you follow this order:

1st + OR -

2nd x OR

3rd OR

Example

Rearrange the formula below to make t the subject.

First move u to the other side to leave the time (t) with the other letter a, we’re trying to make t the subject to do this we’ll have to make sure that t is on its own.

Now to leave t on its own we must divide both sides by a. Now we have;

Lastly we rewrite the equation with the t on the left-hand side

Rearrange the following equations to get the unknown on the left-hand side:

1) 2)

atvv if

3) 4) 5)

6) 7)

8) 9)

10)

11) 12)

13) 14)

2

2

1attvs i

15) 16)

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SET B - SIGNIFICANT FIGURES

Key Concepts

Zeros shown merely to locate a decimal point are NOT significant figures

eg 0.006

The last zeros in a whole number (not a decimal) are somewhat uncertain eg 6 000 000

Zeros located to the right of a number after decimal points are significant eg 0.00600

Zeros between two numbers are significant eg 200002

To find the number of significant figures in a given number: 1. count all the digits starting at the first non-zero digit on the left 2. for a number written in scientific notation count only the digits in the coefficient. Examples

0.1 (1 s.f.) 0.001 (1 s.f.) 0.100 (3 s.f.) 23 (2 s.f.) 2300 (could be 2 or 4 s.f)

How many significant figures are the following numbers written to?

1) 5 ml

2) 571kg

3) 5.71ms-1

4) 0.1V

5) 0.00023 Ω

6) 1.000 A

7) 3.7 x 105 mm

8) 1.0001s

9) 1.00010 ms

10) 5000 K

11) 1.234 mg

12) 0.00077 km

SET C - ROUNDING

When you perform a calculation you often end up with a whole register of numbers showing on your

calculator. It is not relevant to give an answer as 3.24637298 if the number does not reflect that degree of

precision or the resolution of the measurement. To truncate a number you may need to perform rounding.

If the digit to be dropped in rounding is less than 5, the preceding digit is not changed.

If the digit to be dropped in rounding is 5 or greater, the preceding digit is raised by 1.

Write 7.84625 to 1 sig fig 8

Write 7.84625 to 2 sig figs 7.8

Write 7.84625 to 3 sig figs 7.85

Write 7.84625 to 4 sig figs 7.846

Write 7.84625 to 5 sig figs 7.8463

may or may not be

significant

significant significantsignif

isignificant

not significant

significant

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Round to 3 significant figures

Round to 2 significant figures

1) 6.543 6) 6.543

2) 2.367 7) 2.367

3) 1.254 8) 1.254

4) 5.349 9) 5.349

5) 11.15 10) 11.15

SET D - SCIENTIFIC NOTATION

A number can be converted to scientific notation by increasing the power of ten by one for each place the decimal point is moved to the left.

6 6 0 0 0 0.

6.6 x 105 (2 s.f)

A number smaller than 1 can be converted to scientific notation by decreasing the power of ten by one for each place the decimal point is moved to the right.

0.0 0 0 5

5.0 x 10-4 (2 s.f)

Cases to consider: When you don’t move the decimal place anywhere, the power of ten is zero

1.2

1.2 x 100 (2 s.f)

Cases to consider: Moving the decimal place once is written to the power of one.

12

1.2 x 101 (2 s.f)

Note: The number 6x109 may appear on your calculator as 6e+9.

Write the following in scientific notation to 2 significant figures, rounding may be necessary.

1) 5000

2) 614

3) 615

4) 10

5) 0.0250

6) 1

7) 0.1

8) 561.20

9) .0000001

To change scientific notation into standard notation:

Follow the reverse procedure.

3.7 x 103

3. 7 0 0

3700

6.2 x 10-2

0 0 6.2 0.062

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Write the following in standard notation:

10) 3.0 x 106

11) 3.30 x 101

12) 3.33 x 100

13) 6.6 x 10-1

14) 6.6 x 10-3

15) 6.6 x 10-5

16) 1.2 x 104

17) 3.0 x 10-2

18) 5.0 x 100

SET E - USING PYTHAGORAS THEOREM

In a right angle triangle, the square of the hypotenuse, c, is equal to the sum of the squares of the lengths

of the other two sides, a and b.

Find x

1)

2)

3)

4)

5)

6)

X 7cm

4cm

X

4.2cm 3.7cm

X

6.9m 5.3m

X

17m 22m

X

14cm

22cm

X 6cm

3cm

c

a

b

eg b = 8 c = 10 a = ?

= = = 6

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SET F - TRIGNONOMETRY

The set of trigonometric relationships of right angled triangles can be remembered with SOH CAH TOA

sinopposite

hypotenuse

cosadjacent

hypotenuse

tanopposite

adjacent

Find x. (use 2nd to sin-1)

1)

2)

3)

4)

5)

6)

hypotenuse

adjacent

opposite

θ

cm

30o

x cm

55o

6cm

50o

cm

6cm

4.5cm cm

6cm

4cm

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SET G – REPRESENTING VECTORS Vectors are quantities that have magnitude and direction like displacement, velocity, acceleration, force etc.

Scalars are quantities that have only magnitude (no direction) like speed, mass, time etc. To represent their direction

we often use the orientation to the gravitational field (up, down or some angle to the horizontal) or compass points

(NESW) . Length represents the vectors magnitude.

Orientation to the Gravitational Field

Compass Direction

In preference to NESW use “true north” which is specified as degree true (oT). North is used as a reference direction

and all other directions are defined as being at an angle clockwise to North.

a velocity vector 12ms-1 0oT

an acceleration vector 20ms-2 90oT

a force vector 5N 180oT

a displacement vector 2 m 270oT

Represent the following vectors to scale, use a protractor to measure the angle. Indicate the scale (eg 1m = 1cm).

1) 5m90oT 2) 15ms-1 45oT 3) 3N 57 oT 4) 100m 280 oT

SET H – VECTOR ADDITION AND SUBTRACTION Graphically: Adding two vectors A and B graphically can be visualized like two successive walks, with the

vector sum being the vector distance from the beginning to the end point.

+ =

Add the two vectors on graph paper, using a protractor and ruler and specifying a scale. Give the

magnitude and direction of the resultant vector.

1) 3m 90 oT + 5m 270oT

2) 5m 90oT + 3m 0oT

3) 3N 45oT + 3N 315oT

4) 4N 45oT + 2N 45oT

5) 5ms-1 135oT + 5 ms-1 225oT

6) 15m 30 oT + 5m 110oT

5 ms-1 30o above the horizontal

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Subtracting a vector. When subtracting you simply reverse the direction of the vector and then do an

addition.

- = + =

Add the two vectors on graph paper, using a protractor and ruler and specifying a scale. Give the

magnitude and direction of the resultant vector.

7) 3m 90 oT - 5m 270oT

8) 5m 90oT - 3m 0oT

9) 200m 45 oT - 200m 135oT

10) 5m 90oT - 3m 90oT

SET I - UNIT CONVERSIONS SI BASE UNITS

Base units are a fundamental unit in the SI (Standard International) system.

Base Unit metre kilogram second ampere kelvin

Symbol m kg s A K

Prefix Symbol Value

tera T 1012

giga G 109

mega M 106

kilo k 103

centi c 10–2

milli m 10–3

micro μ 10–6

nano n 10–9

pico p 10–12

Just place the exponent from the table above eg

convert 3km see table: kilo, k, 103

= 3 x 103m

convert 7.2 Gs (3 giga seconds)

= 7.2 x 109s

The rules are a little different for mass, as the base unit is in kilograms. As you divide by 1000 to get g into kg eg 5g = 5/1000kg = 0.005kg You can use the technique above to get the answer in grams, then divide by 1000.

eg 100Gg = 100 x 106g (in grams) = 100 x 106 / 1000 kg (in kilograms)

= 10000kg

Change following into base units

1) 20 km

2) 3.2 mm

3) 300 nm

4) 524 ns

5) 55 µs

6) 100 cs

7) 56 g

8) 92 mg

9) 200 µg

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SET J - SLOPE OF A LINE.

There is 100% certainty that this skill will be tested in practical investigations, the mid-year and end of year

exams.

First some rules

don’t choose points that you have plotted – choose points that are ON THE LINE.

choose points that are far apart. Choosing points close together reduces your accuracy.

in physics the slope of a line usually has units! e.g. ms-1.

What is the slope of the lines

1) AB 2) CD

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3) What is the slope of the line above?

4) What are the appropriate units?

5) What is the slope of the line above?

6) What are the appropriate units?

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SET K - EQUATION OF A STRAIGHT LINE.

The equation for a straight line is of the form y = mx + c

Where m is the slope and c is the y intercept.

For the graph below.

Find the slope m using the equation,

Using the points A and B, A is (-6,0) and B is (3,6)

=

= 0.7 (1s.f.)

The y intercept occurs at C, c = 4

Therefore the equation of the line is

y = 0.7 x + 4

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Find the y intercept for lines:

1) AB 2) CD 3) EF

Find the slope for lines:

4) AB 5) CD 6) EF

Write the equations for lines:

7) AB 8) CD 9) EF

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SET L - INTERPRETING GRAPHS. (PROPORTIONAL RELATIONSHIPS)

Below are the graphs of y = x and y = x + 2

Whenever a graph of y against x produces a straight

line then it suggests that

Below are the graphs of y = x2 ,

and

Whenever a graph of y against x produces a parabolic shape in the first quadrant it suggests the

possibility that

To test which relationship may be supported successively draw graphs of y against xn, starting with n=2 i.e.

graph y against x2 then y against x3 etc. until a straight line is produced. When a straight line is produced it

supports the suggestion that for the appropriate value of n.

To the right are the hyperbolic shaped graphs of y = 1/x, y = 1/x2 and 1/x3

Whenever a graph of y against x produces graphs

similar to those above it suggests the possibility that

To test which relationship may be supported

successively draw graphs of y against 1/xn, starting

with n=1 i.e. graph y against 1/x , then graph y

against 1/x2 etc. until a straight line is produced.

When a straight line is produced it supports the

suggestion that

for the appropriate value

of n.

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Section No. of

Problems Student Comment

SET A - Rearranging Equations 16

SET B - Significant figures 12

SET C - Rounding 10

SET D - Scientific notation 9

SET E - Using Pythagoras Theorem 6

SET F - Trignonometry 6

SET G – Representing Vectors 4

SET H – Vector Addition and Subtraction

10

SET I - Unit Conversions SI base units 9

SET J - Slope of a line. 6

SET K - Equation of a straight line. 9