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Physics Factsheet

September 2000

Number 03

Maths for Physics: Algebraic Manipulation I Algebra is used frequently in almost every area of Physics. This Factsheet reviews the key algebraic techniques and concepts required as well as providing strategies for solving equations effectively and avoiding common mistakes. What Is Algebra and Why Do We Use

What Is Algebra?onships between various quantities like mass, Word Relationships between various quantities (like mass, volume and density) are best expressed in terms of some formula or equation (an equation is anything with an = sign in it): Word Equation:

mass density = volume

Symbol Equation:

m ρ= V

Example:

Example: 3z2y4yz simplifies to 12z3y2, as this is: 3× z× z× y× 4× y× z = 3× 4× z× z× z× y× y II. Multiplying Out Brackets 1. One bracket and a number/letter Multiply the term outside (including sign) by each of the terms inside (including sign). Remember: w signs "belong to" the term after them w if there isn't a sign written in, it's +

ρ: density m: mass V: volume

We usually use symbol equations because they are much quicker to write! Algebra is just a way of manipulating numbers without actually saying what the particular numbers are; we can think of the letters in the formula as slots into which we can put particular numbers to work out another quantity.

Example: 2y(y − 4) = 2y× y + 2y× (-4) = 2y2 − 8y Example: -3a(4a − 2b) = (-3a)× 4a + (-3a)× (-2b) = -12a2 + 6ab

Example: Calculate the density of glass, given that the mass of 2.45m3 is 5978kg. We slot the numbers we know into the formula:

ρ

.

• • •

=

y2 yy y simplifies to y, as this is really y

2. Two brackets An easy way to do this is to use FOIL, which stands for First, Outer, Inner, Last. Remember to take account of signs!

m

V 5978 = 2.45 = 2440 kgm-3

Example: Expand (y − 3)(y + 2) First: y× y = y2 Outer: y× (+2) = 2y Inner: (-3)× y = -3y Last: (-3)× (+2) = -6 Now we combine the terms and simplify: y2 + 2y − 3y − 6 = y2 − y − 6

To avoid getting in a mess with algebra, remember the following: All operators (× ,÷,+,−,√) have the same functions in algebra as they do when used with numbers. If letters are written next to each other, it means they are multiplied. eg 2as means 2 × a ×s

III. Factorising This is putting things back into brackets. We will only be looking at factorising involving one bracket here. This is one method of doing it:

BIDMAS - stands for Brackets, Index (= power), Divide,, Multiply, Add, Subtract. This tells you in what order to do operations. eg. 2 + 3a2 means first square a, then multiply by 3, then add 2

Step 1: Look for a factor of both the numbers - that's something that goes into both of them. Write downwhat it is, then cancel it from the numbers. Step 2: Repeat if necessary until there is nothing left that goes into both numbers. Step 3: Look for a factor of both the letter terms. Write down what it is, then cancel it from the letters, and repeat if necessary. One way to do this is to expand the letter parts of the terms out, so y2 becomes y× y, and cross out letters that match in the two terms. Step 4: Put the factors outside the brackets. What's left after you've done all the cancelling goes inside.

Basic Techniques In Physics, you will need algebra to rearrange, solve and substitute into equations. In order to do this, there are a few basic techniques to revise.

I. Simplifying 1. Addition and Subtraction Identical groups of symbols separated by ‘+’ or ‘-‘ can be combined. Example : Can we simplify E = V + Ir? No – there are no identical combinations of symbols. Example: Can we simplify E = V + Ir + 2Ir? Yes – there are identical combinations of symbols (Ir) , E = V + 3Ir Example: Can we simplify y = x + x2 ? No – there are no identical combinations of symbols (they have different powers).

Example: Factorise 4y2 − 6y 2 is a factor of both 4 and 6.

Cancelling it, we get 2y2 - 3y

2y2 - 3y can be written as 2× y× y − 3× y We have a "matching" y in both terms, so y is a factor. Cancelling it, we get 2× y× y − 3× y = 2y − 3

2. Multiplication and Division Identical terms on thetop and bottom of fractions can be cancelled.

So we put 2 y (from the factors 2 and y) outside the bracket, and 2y − 3 inside: 4y2 − 6y = 2y(2y− 3)

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Physics Factsheet

Maths for Physics I - Algebraic Manipulation Example: Factorise 9ar2 + 3r2 3 is a factor of 9 and 3. Cancelling: 3ar2 + r2 3ar2 + r2 can be written as 3× a× r2+ r2 So r2 is a factor of both. Tip: We could expand the r 2 as Cancelling it, we get 3× a× r2+ r2 = 3a + 1 So we get: 3r2 (3a + 1)

Example 2:

We need to get rid of the v2 first The opposite of " + v2" is "−v2". So we subtract v2 from both sides -2as = u2− v2

well, but we don't need to as it's in both. Tip: When we cancel out everything in the last term, we are left with 1, not 0, because we are dividing:- r2÷ r2 = 1

number after the "a" in the same way as one before it.

Rearranging Equations

We can simplify this by multiplying top and bottom by -1 (which

m If we wanted to use the equation ρ = V to find a mass or a volume instead of a density, we'd need to rearrange it. There are two ways of approaching rearranging equations:

Put in the numbers then rearrange

2

changes ALL the signs): a =

or

•

v - u 2s

2

V Example 3: Make I the subject of the equation R = I

Rearrange first then put in the numbers

We need to get rid of the denominator first - so multiply by I: RI = V Tip: You could have written IR

If you find letters very off-putting, you may prefer to do it the first way. But if you have a lot of the same sort of calculation to do, or if you are prone to making mistakes when writing out large numbers, then you're better off with the second method. In either case, you need to follow the same approach and remember a few simple rules. The general principals to remember when rearranging are:

•

Tip: v2 - 2as is NOT THE SAME as 2as - v2 . The minus sign "belongs" to what comes after it - the 2as. The v2 is positive

Now we need to get rid of the"-2s" in front of the a. So we divide the whole equation by "-2s" 2 2 u -v Tip: "2as" and "2sa" are the same as a = -2s each other. You get rid of a letter or

Tip: If you aren't sure, multiply out again to check!

• •

Make a the subject of v2 − 2as = u2

instead - it's just the same.

Now we get I on its own - so divide by R: V I= R

You must ALWAYS do the same thing to BOTH sides of the equation. To get rid of something, you need to do its opposite eg to get rid of + 4 , you must do its opposite, that is −4.

Exam Hints - Quick Methods Triangles You can use a triangle to help you with any equation that can be written as: something = two things multiplied together (like V = IR or m=ρV).

1. Basic Method To rearrange an equation, follow these steps: First simplify it, if possible. Then, focusing on the side of the equation containing the letter you want to find: 1. Get rid of any pluses or minuses that are not part of fractions 2. Get rid of any denominators (bottoms) of fractions by multiplying through by them - even if they include the letter you want to find (careful! the WHOLE of the equation must be multiplied - you may need to use brackets!) 3. Get rid of any pluses or minuses that are left 4. Get rid of any numbers/letters in front of the one you want by dividing by them (again - the whole equation must be divided)

Put the letter on its own in the top of the triangle, and the other two letters underneath:

V I× × R Then you cover up the letter you want to find; what's left shows you the formula.

Example: To find R, cover it up. We are then left with

V I

. So R =

V I

Example 1: Make t the subject of the equation: v = u + at. We must do something about the + first We want to keep the "at" (since it's "t" we want) so we need to get rid of the u that's added to it.

You may also see this sort of equation written as something = fraction; whatever's on the top of the fraction goes in the top of the triangle

Tip: u + at means the same as at + u. Rewrite it this way if you find it easier!

Cross-Multiplying For equations that look like one fraction = another fraction, you can save some time by multiplying the top of each fraction by the bottom of the other:

The opposite of "+ u" is " − u". So we subtract u from both sides: v − u = at

a c = becomes ad = bc (check this). b d

Now we must get rid of the "a" in front of the "t". We divide the whole equation by a: Tip: NOT v-u/a = t. v-u t= a Both the v and the u must be divided by a.

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Physics Factsheet

Maths for Physics I - Algebraic Manipulation

Example 1:

2. Equations with Squares and Square Roots If the letter that you want to make the subject of the equation is squared or square rooted, you need to do the following: 1. Get the part that is squared/ square-rooted on its own (careful! ax2 means just the x is squared). 2. Get rid of the square/ square-root by doing the opposite to the whole of both sides of the equation 3. Then rearrange as necessary. Example 1: Make m the subject of the equation: ω =

Make x the subject of ax + b = d - fx

First we get all the terms containing x onto one side - we add fx to both sides: ax + b + fx = d

Tip: It's usually easier to get the x's onto the side where they are positive.

Now we get rid of the terms not involving x from the left hand side we subtract b from both sides: ax + fx = d − b

k m

Now we take x out as a common factor on the left hand side: x(a + f) = d − b.

We already have the square-rooted part on its own, so we now do the opposite of square-rooting - we square both sides: k 2 ω = m Multiply by m to get rid of the denominator: mω2 = k Divide by ω2 to get m on its own: k m= 2 ω

Now we divide by the bracket next to the x: d −b Tip: Do NOT try to simplify algebraic x= a+ f fractions unless there is a common factor on the top & bottom.

Example 2: Make u the subject of f =

Example 2: Make u the subject of the equation: v2 = u2 + 2as First get the squared part on its own: v2 − 2as= u2

uv u+v

Tip: We can't subtract the v, since it's in a fraction.

We need to get rid of the denominator on the right hand side first, so we multiply by (u + v): Tip: Those brackets must be f(u + v) = uv there, since we are multiplying by

Now we do the opposite of squaring - we square -root both sides: √ (v2 − 2as) = u

the whole of u+v

Tip: This is NOT THE SAME as √v2 − √(2as)

In order to get the u terms together, we'll need to multiply out the brackets: Tip: ONLY multiply out fu + fv = uv brackets if the letter you are looking for is inside them.

Now we can get the u terms on one side: fv = uv − fu

3. Equations where the letter you want is in two places In equations like this, you need to do the following:

Factorise: fv = u(v − f)

1. Get all the terms involving the letter you want onto one side, and the other terms on the other side. 2. Take out the letter you want as a factor 3. Divide by the bracket next to the letter you want.

Divide: u=

fv v− f

Table 1. Some common errors in algebra The commonest algebraic errors arise because of fear of fractions and brackets. If you stick to the above rules (don’t invent any of your own!), and do the calculation in steps rather than all at once, you should be able to avoid mistakes. Here are some of the most common errors that students make: Wrong version

û

ü

Correct version

Comments

a(b+c) = ab + c

a(b+c) = ab + ac

The a multiplies both b & c

a - (b - c) = a - b - c

a - (b - c) = a - b + c

The - (ie -1) multiplies both b and c

aX + bX = (a + b)X2

aX + bX = (a + b)X

Having two X’s doesn’t mean you have to multiply them to get X2...

aX + bX = 2X(a + b)

aX + bX = (a + b)X

... nor do you add them to get 2X as if the a and the b were irrelevant.

2 A = π ( d )2 = π d

( d2 )

Area of circle: A = π ( d ) 2 2

= π d2/2 ab + c = d ⇒ a = d/b - c

2

4

ab + c = d ⇒ a = (d - c)/b

2

2 means d × d , which is d × d , equal to d 2 2 2×2 4

You shouldn’t make this mistake if you do the calculation in steps: ab + c = d ⇒ ab = d - c (subtract c from both sides) ⇒ a = (d - c)/b (divide both sides by b)

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Physics Factsheet

Maths for Physics I - Algebraic Manipulation

rL 5. The formula R = A relates the resistance of a length of wire (R),

Solving Basic Equations Solving equations involves finding the value of a letter. The method is very similar to rearranging equations - you are making the letter you want to find the subject of the equation.

its resistivity (ρ), its length (L metres) and its cross-sectional area (A metres2).

Example 1. Solve the equation 8 + 2x = 6 − 3x

a) Rearrange this equation into a suitable form for calculating resistivity.

Since x occurs in two places, we need to get all the x terms on one side, and everything else on the other side: Add 3x: 8 + 2x + 3x = 6

Solutions

Subtract 8: 2x + 3x = 6 − 8

g)

Divide by 5: x = -0.4

r Solve the equation 4 = r

2

Cross multiply: 4× 2 = r× r

b) 3x2 – 4x c) -8y – 20y2 2 e) x – x - 6 f) 2x2 – 19x + 35 h) 2x3 + 5x2 + 4x + 10

3. a) 2(3x – 2) d) 7a(b – 3) g) ac(b – d)

b) x(2x – 5) e) πr(r + 2L) h) 8x3(3 – 4x)

d) To get rid of the square, square- root: r =√ 8=2. 83 (3 SF)

t=

2 b) m = Fr GM gT2 e) t = 2 4p

Q I

c) f)

GMm F 2 - 3y x= y +1

r=

h) v = √(u2 + 2as)

R 2R R2 - R

RA L rL b) A = R A = πr2

c) 2x(3 – 2x) f) 4ab(2a + 3b)

5. a) r =

1. Simplify the following as far as possible a) 3ab + 2a – 3b – ab b) 2x2 + 3x – 4x + 5 + 6x2 c) 2a × 3a d) e2 × 4e3

g)

1 u = s - at t 2

g) R 1 =

Questions

6ab 2ac

3b c

c) 6a2 f) 2x

2. a) 6a – 3ab d) -12a2b + 3a3 g) -28x2 – x + 2

4. a)

Simplify: 8 = r2

e) 3ab × 2a

b) 8x2 – x + 5 e) 6a2b a+b h) c-b

1. a) 2ab + 2a – 3b d) 4e5

Simplify: 5x = - 2

Example 2:

b) A piece of wire has resistivity 0.00024 Ωm, resistance 6Ω and length 0.5 metres Calculate its radius.

f) h)

2x 2

⇒

A = 0.000002 m2

0.000002 = πr2

⇒ r = 7.98 × 10-4 m

x a+b c-b

Related "Maths for Physicists" Factsheets

• • • • • • • • •

2. Multiply out the following brackets, and simplify the answer as far as possible a) 3a(2 – b) b) x(3x – 4) c) -4y(2 + 5y) d) -3a2(4b – a) e) (x + 2)(x − 3) f) (2x – 5)(x – 7) g) (1 – 4x)(2 + 7x) h) (x2 + 2)(2x + 5) 3. Factorise the following a) 6x – 4 b) 2x2 – 5x d) 7ab – 21a e) πr2 + 2πrL g) abc – cad h) 24x3 - 32x4

⇒

c) 6x – 4x2 f) 8a2b + 12ab2

4. Rearrange each of the following to make the letter indicated the subject GMm u b) F = m a) s = ut + ½ at2 r2 Q GMm c) F = r d) I = t t r2 L 2-x L f) y = e) T = 2p g x 3+ x 1 1 1 g) R = R 1 + R 2 R1 h) 2as = v2 – u2 v

Numerical Calculations & Data Handling Numerical Calculations & Units Indices Exponentials and Logs Further Algebraic Manipulation Geometry Graphs 1 Graphs 2 Graphs 3

Further Practice • Any GCSE Mathematics Higher Tier textbook will provide further practice on rearranging and solving equations. Acknowledgements: This Physics Factsheet was researched and written by Cath Brown The Curriculum Press,Unit 305B, The Big Peg,120 Vyse Street, Birmingham, B18 6NF Physics Factsheets may be copied free of charge by teaching staff or students, provided that their school is a registered subscriber. No part of these Factsheets may be reproduced, stored in a retrieval system, or transmitted, in any other form or by any other means, without the prior permission of the publisher. ISSN 1351-5136

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