PHYSICS MATH SKILLS BOOKLET 2012
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. Self-evaluation The problems too hard and I couldn’t do The problems too hard them – I need and I couldn’t do HELP! them –
Self-evaluation The problems were The problems were more hard than easy more easy than hard The problems were more hard than easy
The problems were more easy than hard
No problemo – too easy! No problemo – too easy!
I need HELP!
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
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Set L - Interpreting graphs. (Proportional relationships) ................................................................................. 13
1
SET K - Equation of a straight line. ................................................................................................................... 11
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 2nd 3rd
+ OR x OR 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: v f vi at
1) 2)
3) 4) 5)
6) 7)
8) 9)
10)
11)
1 s vi t at 2 2 15) 16)
2
12)
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13) 14)
SET B - SIGNIFICANT FIGURES Key Concepts Zeros shown merely to locate a decimal point are NOT significant figures not significant
The last zeros in a whole number (not a decimal) are somewhat uncertain may or may not be significant
eg 6 000 000
eg 0.006
Zeros located to the right of a number after decimal Zeros between two numbers are significant points are significant significantsignif significant significant isignificant
eg 0.00600
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? ď‚Ť 1) 2) 3) 4)
5 ml 571kg 5.71ms-1 0.1V
5) 6) 7) 8)
0.00023 â„Ś 1.000 A 3.7 x 105 mm 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.
3
Write 7.84625 to 2 sig figs 7.8 Write 7.84625 to 4 sig figs 7.846
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Write 7.84625 to 1 sig fig 8 Write 7.84625 to 3 sig figs 7.85 Write 7.84625 to 5 sig figs 7.8463
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 A number smaller than 1 can be converted to increasing the power of ten by one for each place scientific notation by decreasing the power of ten the decimal point is moved to the left. by one for each place the decimal point is moved to the right.
6 6 0 0 0 0. 6.6 x 105
0.0 0 0 5 5.0 x 10-4 (2 s.f)
(2 s.f)
Cases to consider: Cases to consider: When you don’t move the decimal place anywhere, Moving the decimal place once is written to the the power of ten is zero power of one.
1.2
12
1.2 x 100 (2 s.f)
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:
3. 7 0 0 3700
0 0 6.2 0.062
4
3.7 x 103
6.2 x 10-2
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Follow the reverse procedure.
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. eg b = 8 c = 10 a = ?
c
b =
a
=
= 6
Find x
1)
6cm
X
3)
5)
3cm
6.9m
4.2cm
3.7cm
5.3m
X X 14cm
17m
X 22m
7cm
4cm
5
X 22cm
6)
X
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2)
4)
SET F - TRIGNONOMETRY The set of trigonometric relationships of right angled triangles can be remembered with SOH CAH TOA
sin
opposite hypotenuse
cos
adjacent hypotenuse
tan
opposite adjacent
θ
hypotenuse
adjacent opposite
(use 2nd to sin-1)
Find x.
1)
4) 6cm cm
cm 30o
2)
5) cm
cm
4.5cm
x 55 o
3) 50o
6) 4cm
6
6cm
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6cm
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 5 ms-1 30o above the horizontal
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
an acceleration vector
a force vector
a displacement vector
12ms-1 0oT
20ms-2 90oT
5N 180oT
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.
7
4) 4N 45oT + 2N 45oT 5) 5ms-1 135oT + 5 ms-1 225oT 6) 15m 30 oT + 5m 110oT
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1) 3m 90 oT + 5m 270oT 2) 5m 90oT + 3m 0oT 3) 3N 45oT + 3N 315oT
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 Symbol
metre m
kilogram kg Prefix tera giga mega kilo centi milli micro nano pico
Symbol T G M k c m μ n p
second s
ampere A
kelvin K
Value 1012 109 106 103 10–2 10–3 10–6 10–9 10–12
Just place the exponent from the table above eg convert 7.2 Gs (3 giga seconds) 3 convert 3km see table: kilo, k, 10 9 3 = 7.2 x 10 s = 3 x 10 m The rules are a little different for mass, as the base eg 100Gg unit is in kilograms. As you divide by 1000 to get g = 100 x 106g (in grams) into kg eg 5g = 5/1000kg = 0.005kg = 100 x 106 / 1000 kg (in kilograms) You can use the technique above to get the answer = 10000kg in grams, then divide by 1000.
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7) 56 g 8) 92 mg 9) 200 µg
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Change following into base units 1) 20 km 4) 524 ns 2) 3.2 mm 5) 55 µs 3) 300 nm 6) 100 cs
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
What is the slope of the lines 1) AB
2) CD
9
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.
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3) What is the slope of the line above? 4) What are the appropriate units?
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10
5) What is the slope of the line above? 6) What are the appropriate units?
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
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11
y = 0.7 x + 4
Find the y intercept for lines: 1) AB
5) CD
6) EF
Write the equations for lines: 8) CD
9) EF
12
4) AB
7) AB
3) EF
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Find the slope for lines:
2) CD
SET L - INTERPRETING GRAPHS . (PROPORTIONAL RELATIONSHIPS) Below are the graphs of y = x and y = x + 2
Below are the graphs of y = x2 ,
and
Whenever a graph of y against x produces a straight Whenever a graph of y against x produces a parabolic shape in the first quadrant it suggests the line then it suggests that 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
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13
of n.
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
Student Comment
14
No. of Problems
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Section