# A2 Singles 2021 Learningpack term 1 ```BHASVIC Maths
Y2 Learning Pack
September – October 2021
0
A LEVEL MATHEMATICS YEAR 2 LEARNING PACK CONTENTS
Your teacher will tell you which lesson to watch the videos by. If they don’t then assume all
videos need to be watched by the first lesson of the week.
Week
-1
4th Sept
0
13th Sept
1
20th Sept
2
27th
3
Sept
4th Oct
Ref
in
spec
P5
P5
P5
P5
P2
P2
P2
P2
P2
P2
P8
P8
P8
P5
P5
P5
P7
P1
P1
P1
P1
P1
P5
P5
P7
P7
Videos
Page
Watch
by
lesson
Prior Knowledge – Compound Angles
Compound Angle Proofs and Solves
Double Angle Proofs and Solves
Trig Proofs
Prior Knowledge -Partial Fractions
Partial Fractions – 2 linear factors
Partial Fractions – 3 linear factors
Partial Fractions – Repeated Root
Partial Fractions – Alternative Method
Partial Fractions – Top Heavy
Integration Using Partial Fractions
Prior Knowledge - Integration using Trig
Identities
Integration using Trig Identities
Prior Knowledge -Small Angle Approximations
Small Angle Approximations - proof
Small Angle Approximations - applications
Differentiation of sin x and cos x from 1st
principles
Prior Knowledge - Proof
Proof by Deduction
Proof by Exhaustion
Proof by Counter Example
Prior Knowledge - Inverse Trig – graphs and
equations
Inverse Trig – graphs and equations
Prior Knowledge - Implicit differentiation
Implicit Differentiation
3
4
7
9
10
11
12
13
13
14
15
16
Mini lesson
17
23
24
25
26
1 2 3
1 2 3
27
28
29
30
31
32
1
1
1
1
1
1
33
38
39
1 2 3
1 2 3
1 2 3
1
Mini lesson
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
1 2 3
1 2 3
2
2
2
2
2
2
3
3
3
3
3
3
Tick
when
watched
4
11th Oct
P2
P2
P2
P2
P2
5
18th Oct
SM9
SM9
SM9
SM9
25TH Oct
Prior Knowledge - Binomial Expansion
Binomial expansion - Introduction
Finite Binomial Expansion where n is a positive
integer
Infinite Binomial Expansion where n
is fractional or negative
Binomial Expansion using Partial Fractions
43
44
45
1 2 3
1 2 3
1 2 3
46
1 2 3
48
1 2 3
Prior Knowledge - Horizontal Moments
Horizontal Moments - Introduction
Horizontal Moments - Beams
Horizontal Moments - Tilting
HALF TERM
49
50
52
53
1
1
1
1
2
2
2
2
3
3
3
3
Reference to specification
Reference to
specification
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
Topic
Reference to
specification
SM1
SM2
SM3
SM4
SM5
SM6
SM7
SM8
SM9
SM1
Proof
Algebra and functions
Coordinate geometry in the (x,y) plane
Sequences and series
Trigonometry
Exponentials and logarithms
Differentiation
Integration
Numerical methods
Vectors
2
Topic
Sampling
Data Presentation and interpretation
Probability
Statistical distributions
Statistical hypothesis testing
Quantities and units in mechanics
Kinematics
Forces and Newton’s laws
Moments
Sampling
Compound Angle Prior Knowledge Work
1. Find the exact values of:
(a) sin 45&deg; (b) cos
𝜋
(c) tan
6
𝜋
3
2. Solve the following equations in the interval 0 ≤ 𝑥 ≤ 360&deg;.
(a) sin(𝑥 + 50&deg;) = −0.9
(b) cos(2𝑥 − 30&deg;) =
1
2
(c) 2 sin2 𝑥 − sin 𝑥 − 3 = 0
3. Prove the following:
(a) cos 𝑥 + sin 𝑥 tan 𝑥 ≡ sec 𝑥
(b) cot 𝑥 sec 𝑥 sin 𝑥 ≡ 1
(c)
cos2 𝑥+sin2 𝑥
1+cot2 𝑥
≡ sec 2 𝑥
1.
(a)
1
√2
(b)
√3
2
(c) √3
2.
(a) 194.2&deg;, 245.8&deg;
(b) 45&deg;, 165&deg;, 225&deg;, 345&deg;
(c) 270&deg;
3.
Proofs
3
P5 – Trigonometry
https://drive.explaineverything.com/thecode/AMLHFXQ
What is sin (10 + 20)?
Why?
4
22min
In the formula book
sin (A+B) =
cos (A+B) =
sin (A-B) =
cos (A-B) =
tan (A+B) =
tan (A-B) =
The Proof of tan (A+B) (this needs learning off by heart)
Prove that tan (A+B) =
5
Example 1
Express the following as a single sine, cosine or tangent:
𝑡𝑎𝑛 60 − 𝑡𝑎𝑛 45
1 + 𝑡𝑎𝑛 60 𝑡𝑎𝑛45
Example 2
Calculate the exact value of tan 𝑥 𝑓𝑜𝑟
1
sin(𝑥 − 30) = cos 𝑥
2
Example 3
6
P5 – Trigonometry
The Double Angle formulae
https://drive.explaineverything.com/thecode/CGETHCB
In the formula book
The Double angle formulae are:
sin (2A) =
cos (2A) =
cos (2A) =
cos (2A) =
tan (2A) =
7
22 min
Example 1
Solve 5 𝑠𝑖𝑛2𝜃 + 4 𝑠𝑖𝑛𝜃 = 0 𝑓𝑜𝑟 0 ≤ 𝜃 ≤ 360
Example 2
Solve 3 cos 2𝑥 − cos 𝑥 + 2 = 0 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 360
Something clever (manipulating the formulae):
Example 3
Solve sin 4 𝜃 = 𝑐𝑜𝑠 2𝜃 𝑓𝑜𝑟 0 ≤ 𝜃 ≤ 𝜋
Example 4
𝜃
Solve 3𝑐𝑜𝑠𝜃 − 𝑠𝑖𝑛 (2 ) −1 = 0 𝑓𝑜𝑟 0 ≤ 𝜃 ≤ 𝜋
8
P5 – Trigonometry
Trig proofs
https://drive.explaineverything.com/thecode/BLKLKSX
Example 1
cos 2𝐴
Prove 𝑐𝑜𝑠2 𝐴−𝑐𝑜𝑠𝐴𝑠𝑖𝑛𝐴 ≡ 1 + tan 𝐴
Example 2
Prove
𝑐𝑜𝑠3𝜃
𝑠𝑖𝑛𝜃
+
𝑠𝑖𝑛3𝜃
𝑐𝑜𝑠𝜃
≡ 2 cot 2𝜃
Example 3
Prove cos 3𝐴 ≡ 4𝑐𝑜𝑠 3 𝐴 − 3 cos 𝐴
9
10min
Partial Fractions Prior Knowledge Work:
To prepare for this week’s topic, you need a sound understanding of algebraic fractions and algebraic
manipulation.
Try these questions and seek out help if you need it from your teacher or other students.
1. Write each of this expression as a single fraction.
4
3
2
+
+
𝑥 + 2 𝑥 + 3 (𝑥 + 2)2
https://youtu.be/NvdT5wKP160
2. Integrate the following expressions:
3
(a) ∫ 3𝑥−1 𝑑𝑥
6
𝑥
(b) ∫ 2𝑥−5 𝑑𝑥
(c) ∫ 𝑥 2 +4 𝑑𝑥
2.
(a) ln(3𝑥 − 1) + 𝑐
(b) 3 ln(2𝑥 − 5) + 𝑐
10
1
(d) 2 ln(𝑥 2 + 4) + 𝑐
P1 – Algebra and Functions
1. Partial Fractions – 2 Linear Factors
https://youtu.be/NUuJlgmb8PA
3 mins
𝑥+3
(𝑥 + 1)(𝑥 + 2)
Now you try this:
11
P1- Algebra and Functions
2. Partial Fractions – 3 Linear Factors
https://youtu.be/dCmrPqIfWW0
5 mins
5𝑥 2 + 3𝑥 + 4
(𝑥 + 1)(2𝑥 + 1)(𝑥 − 2)
Now you try this:
12
P1 – Algebra and Functions
3. Partial Fractions – Repeated Factors
6 mins
https://youtu.be/qztRKnUnuC4
4𝑥 2 − 3𝑥 + 2
(𝑥 + 1)(𝑥 − 2)2
4. Alternative Method
https://youtu.be/HDzfnWelx3o
10 mins
2𝑥 2 − 3
(𝑥 − 2)(𝑥 + 3)2
13
P1 Algebra and Functions
5. Partial Fractions – Top Heavy
https://youtu.be/SO8GfX9taJA
4 mins
2𝑥 2 + 5
(𝑥 + 4)(𝑥 − 2)
14
P8- Integration using Partial Fractions
6. Partial Fractions – Use of Partial Fractions in Integration
https://estream.bhasvic.ac.uk/View.aspx?id=15928~5m~eYAyXjcqPj 4 mins
∫
𝑥2
3𝑥
𝑑𝑥
− 4𝑥 − 5
15
Prior Knowledge: Integration Using Trig Identities
To prepare for this week’s topic, you need to remember the trig identities that we have met thus far
in the course.
You also need to be able to integrate trig functions using the reverse chain rule (GDA).
1. Match up the following trig identities.
2 cos2 𝐴 − 1
sin 𝑥
cos 𝑥
sec 2 𝑥
sin2 𝑥 + cos 2 𝑥
tan 𝐴 &plusmn; tan 𝐵
1 ∓ tan 𝐴 tan 𝐵
tan2 𝑥 + 1
sin 𝐴 cos 𝐵 &plusmn; cos 𝐴 sin 𝐵
1 + cot 2 𝑥
cos2 𝐴 − sin2 𝐴
sin(𝐴 &plusmn; 𝐵)
tan 𝑥
cos(𝐴 &plusmn; 𝐵)
cos 𝐴 cos 𝐵 ∓ sin 𝐴 sin 𝐵
tan(𝐴 &plusmn; 𝐵)
2 sin 𝐴 cos 𝐴
sin 2𝐴
1
cos 2𝐴
2 tan 𝐴
1 − tan2 𝐴
tan 2𝐴
cosec 2 𝑥
1 − 2 sin2 𝐴
2. Integrate the following functions:
𝑥
(b) ∫ sec 2 2 𝑑𝑥
(a) ∫ sin 5𝑥 𝑑𝑥
𝜋
𝑥
3. Evaluate ∫0 cos 2 𝑑𝑥
1
2. (a) 5 cos 5𝑥 + 𝑐
𝑥
(b) 2 tan 2 + 𝑐
3. 2
16
P8 – Integration – Integration Using Trig Identities
Note that the following standard integrals, that can be found in the
differentiation and/or integration sections of the formula booklet, are regularly
used in the integrals in this section.
1. https://youtu.be/mkgOwMD140Y
∫
3 mins
sin 2𝑥
𝑑𝑥
cos 𝑥
17
2.
https://youtu.be/3RIxvN-7IFI
5 mins
∫ tan2 𝑥 + 1 𝑑𝑥
∫ 1 + tan2 5𝜃 𝑑𝜃
∫ 3 + 3 tan2 𝑑𝑥 𝑑𝑥
18
3. https://youtu.be/yc8Kepa1s20
5 mins
∫ sin 𝑥 cos 𝑥 𝑑𝑥
∫ 5 cos
3𝑥
3𝑥
sin 𝑑𝑥
2
2
19
4. https://youtu.be/KrWPb6JTVRc
7 mins
∫ sin2 𝑥 𝑑𝑥
∫ 3 sin2 5𝜃 𝑑𝜃
5. https://youtu.be/7cT0lHvE-fM
7 mins
∫ cos 2 𝑥 𝑑𝑥
∫ 5 cos 2 8𝜃 𝑑𝜃
20
6. https://drive.explaineverything.com/thecode/RKVAWPF
14 mins
∫ 𝑠𝑖𝑛2 3𝑥 𝑑𝑥
∫ 𝑐𝑜𝑠 4 𝑥 𝑑𝑥
To integrate an
even power of sin or cos I must use………
21
∫ 𝑐𝑜𝑠 5 𝑥 𝑑𝑥
To integrate an
odd power of sin or cos I must use………
22
Prior Knowledge Work: Small Angle Approximations and the Differentiation of sin
x and cos x:
To prepare for this week’s topic, we are going to explore the trig graphs close to the y axis, ie. when x
or θ is very small.
1. Use DESMOS (making sure it is in radians) to plot the graphs of y = sin x and y = x.
Zoom in towards the origin. What do you notice about the two graphs?
What does this tell us about sin x and x when x is very small?
2. Use DESMOS (making sure it is in radians) to plot the graphs of y = tan x and y = x.
Zoom in towards the origin. What do you notice about the two graphs?
What does this tell us about tan x and x when x is very small?
3. Use DESMOS (making sure it is in radians) to plot the graphs of y = cos x and y = 1 −
Zoom in towards the point (0, 1). What do you notice about the two graphs?
What does this tell us about cos x and 1 −
𝑥2
2
when x is very small?
You also need to remember how to differentiate from first principals.
4. Prove, from first principals, that the derivative of 4x2 is 8x.
23
𝑥2
.
2
P5 - Trigonometry
1. Small Angle Approximations - Proof
https://youtu.be/-IRHJmBPWw4
11 mins
The small angle approximations are:
sin 𝜃 ≈
cos 𝜃 ≈
tan 𝜃 ≈
This is a really nice proof of where these approximations come from. Try to
follow the proof and take notes. However, you will not be asked for this proof in
an exam.
24
P5 - Trigonometry
1. Small Angle Approximations - Applications
https://youtu.be/ozZINFNndiY
5 mins
Try this question:
25
P7 - Differentiation
1. Proof of the Derivative of sin x and cos x
https://youtu.be/6D1Zrzgq4t4
lim
sin ℎ
ℎ→0
ℎ
10 mins
cos ℎ − 1
=
ℎ→0
ℎ
=
lim
𝑦 = sin 𝑥
𝑦 = 𝑐𝑜𝑠 𝑥
26
Prior Knowledge – Proof
To prepare for this week’s topic you need a good understanding of some
of the proofs you have already covered in the course. If you are stuck
refer to Chapter 7.3 (page 174) in the Year 2 Pure textbook
Try these questions:
1. Using completing the square, prove that 𝑛2 − 8𝑛 + 17 is positive for all
values of n
2. Prove that 𝑠𝑖𝑛𝑥𝑠𝑖𝑛2𝑥 + 𝑐𝑜𝑠𝑥𝑐𝑜𝑠2𝑥 ≡ 𝑐𝑜𝑠𝑥
1. 𝑛2 − 8𝑛 + 17 = (𝑛 − 4)2 + 1
(𝑛 − 4)2 ≥ 0 (as a square number is always greater than or equal to zero)
𝑠𝑜 (𝑛 − 4)2 ≥ 1 (which means it is always a positive number)
Therefore 𝑛2 − 8𝑛 + 17 is positive for all values of n
2. 𝐿𝐻𝑆 = 𝑠𝑖𝑛𝑥𝑠𝑖𝑛2𝑥 + 𝑐𝑜𝑠𝑥𝑐𝑜𝑠2𝑥
= 𝑠𝑖𝑛𝑥 (2 𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥) + 𝑐𝑜𝑠𝑥 (2𝑐𝑜𝑠𝑥 2 − 1)
= 2𝑠𝑖𝑛2 𝑥𝑐𝑜𝑠𝑥 + 2𝑐𝑜𝑠 3 𝑥 − 𝑐𝑜𝑠𝑥
= 2 cos 𝑥 (1 − 𝑐𝑜𝑠 2 𝑥) + 2𝑐𝑜𝑠 3 𝑥 − 𝑐𝑜𝑠𝑥
= 2𝑐𝑜𝑠𝑥 − 2𝑐𝑜𝑠 3 𝑥 + 2𝑐𝑜𝑠 3 𝑥 − 𝑐𝑜𝑠𝑥
= 𝑐𝑜𝑠𝑥
= 𝑅𝐻𝑆
𝑄𝐸𝐷
27
P1 – Proof by deduction
Understand and use the structure of mathematical proof, proceeding from
given assumptions through a series of logical steps to a conclusion. This is the
most commonly used method of proof throughout this specification.
For example, you could be asked to prove by deduction, using completion of
the square, that 3n2 − 18n + 30 is positive for all values of n. Other examples
where you would use this type of proof is if you were asked to prove a simple
derivative from first principles or proving results for arithmetic and geometric
series.
8 mins
1. Prove that for any four consecutive integers, the difference between the product of
the last two and the product of the first two of these numbers is equal to their sum.
2. k3 – k is divisible by 6 for all k &gt; 1
28
P1 – Proof by exhaustion
Proof by exhaustion This involves trying all the options. For example, suppose
x and y are odd positive integers less than 7, prove that their sum is divisible
by 2. To prove this we would go through all the options of adding together 2
positive integers, both of which are less than 7, then showing for each of our
sums, it is divisible by 2.
Prove that 53 is a prime n umber
29
6 mins
P1 – Proof by counter example
For example, suppose that we are asked to show that the statement “n2 – n +
1 is a prime number for all values of n” is untrue. To prove this by counterexample we only have to find a single example of n where n2 – n + 1 is not a
prime number.
6 mins
Prove the following are not true. It is sufficient to find one example where they are not true.
1. 4n + 4 is always a multiple of 8
2. If x &gt; y then x/y &gt; 1
3. 𝑝 − 𝑞 ≤ 𝑝2 − 𝑞 2 for all values of p and q
4. 2n2 – 16 + 31 is always positive
30
Proof by contradiction is performed by initially assuming that the given
statement we are asked to prove is not true. Then using logical deduction we
reach a conclusion that contradicts our assumption. Thus we conclude that
the assumption must be incorrect - in which case the initial statement must
be true. Examples of when we use this method of proof is to prove
irrationality of √p where p is prime. We also use it to show that there are an
infinite number of prime numbers. This method is often the one to choose
when the other methods do not lend themselves to easily starting the proof
Prove that √2 is irrational
31
7 mins
Prior Knowledge Work: Inverse Trig Functions
To prepare for this week’s topic, you need a sound understanding of inverse functions, and solving
trig equations.
Try these questions and seek out help if you need it from your teacher or other students.
1. For each function f(x), find the inverse function f-1(x).
(a) 𝑓(𝑥) = (𝑥 + 3)2 − 2, 𝑥 ≥ −3
(b) 𝑓(𝑥) = 𝑥 2 − 7𝑥 + 1
2. The diagram below shows the graph of 𝑦 = 𝑓(𝑥). Add to the diagram the graph of 𝑦 =
𝑓 −1 (𝑥).
𝜋
3. Given that sin 𝑥 = sin 3 , find all values of x where −2𝜋 &lt; 𝑥 &lt; 2𝜋.
1. (a) 𝑓 −1 (𝑥) = −3 + √𝑥 + 2
7
2
(b) 𝑓 −1 (𝑥) = + √𝑥 +
2. Graph reflected in the line y = x
45
4
𝜋 2𝜋
4𝜋
6𝜋
,− 3 ,− 3
3
3. 𝑥 = 3 ,
32
P5- Trigonometry
1. The Inverse Trig Functions
https://youtu.be/ZM52fcPp1yw 8 mins
arcsin x
https://youtu.be/5o2lqjX-p_Q 6 mins
arccos x
https://youtu.be/1A-IbjIIy_I
6 mins
33
arctan x
P5 - Trigonometry
2. Using the Inverse Trig Functions
https://youtu.be/rUpnyYkVPwQ
14 mins
34
P5 - Trigonometry
3. Using the Inverse Trig Functions
https://youtu.be/k7XlnCoBK28
17 mins
35
36
4. Using the Inverse Trig Functions
https://youtu.be/P3owr0cwZbw
13 mins
37
Prior Knowledge - Implicit Differentiation
To prepare for this week’s topic you need a good understanding of
differentiation and equations of tangents and normals. If you are stuck
refer to 12.6 (page 268) in the Pure Year 2 textbook.
Try these questions
1. Differentiate
a) 3𝑥 2 − 5𝑥
b)
2
𝑥
− √𝑥
c) 4𝑥 2 (1 − 𝑥 2 )
2. Find the equation of the tangent to the curve with equation
𝑦 = 8 − 𝑥 2 at the point (3, −1)
38
P7 – Implicit Differentiation
Differentiate simple functions and relations defined implicitly for first derivative
only. The finding of equations of tangents and normals to curves
given implicitly is required.
https://drive.explaineverything.com/thecode/BKCHBXQ
8 mins
Example 1
𝐹𝑖𝑛𝑑
𝑑𝑦
𝑑𝑥
𝑖𝑓 2𝑦 = 4𝑥 2 + 2
Example 2
𝐹𝑖𝑛𝑑
𝑑𝑦
𝑖𝑓 𝑥 2 + 𝑦 2 = 1
𝑑𝑥
So to differentiate implicitly, differentiate everything you can see , left and
𝑑𝑦
right with respect to x. If you differentiate a y multiply it by 𝑑𝑥
39
Example 3
a) Find the coordinates of the stationary points for 𝑥 2 − 4𝑥𝑦 = 𝑦 2 − 20
b) Show that the gradient is not parallel to the y axis for
𝑥 2 − 4𝑥𝑦 = 𝑦 2 − 20
40
Example 4
Find the gradient of the tangent to the curve (𝑥 + 𝑦)2 = 4𝑥 when 𝑥 = 1, 𝑦 &gt; 0
𝑑𝑦
=0
To find the gradient which is parallel to the
x axis use
To find the gradient which is parallel to the
y axis (or infinite) use
41
𝑑𝑥
𝑑𝑥
𝑑𝑦
=0
Example 5
Prove the differential of 𝑦 = 2𝑥 is 2𝑥 ln 2
42
Prior Knowledge - Binomial Expansion
To prepare for this week’s topic you need a good understanding of
expanding brackets and simplifying indices. If you are stuck refer to 1.2
(page 4) and 1.4 (page 9) in the Pure Year 1 textbook.
Try these questions
1. Expand and simplify where possible
a) (2x – 4y)(3x + y)
b) (x – 4y)(2x + y +5)
c) 3x(x – 2y)(2x + 1)
2. Simplify
a) (−2𝑥)4
b) (−3𝑥)−2
2
3
c) (5 𝑥 3 )
3. Simplify
1
a) (3 𝑥)
−2
b) (125𝑥12 )−2
3
9
c) (4 𝑥 )
4 2
1.
a) 6𝑥 2 − 10𝑥𝑦 − 4𝑦 2
b) 2𝑥 2 + 5𝑥 − 7𝑥𝑦 − 4𝑦 2 − 20𝑦
c) 6𝑥 3 + 15𝑥 2 − 3𝑥 2 𝑦 − 18𝑥𝑦 2 − 30𝑥𝑦
1
2.
a) 16𝑥 4
b) 𝑥 −2
c)
3.
a) 9𝑥 −2
b) 5 𝑥 −6
1
c)
9
43
9
25
𝑥6
27
8
𝑥6
P2 – Binomial expansion - Introduction
Understand and use the binomial expansion of (a + bx)n for positive integer n;
the notations n! and nCr
Use of Pascal’s triangle. Relation between binomial coefficients.
𝑛
Also be aware of alternative notations such as( )
𝑟
https://youtu.be/qZzSWz27ROY
14 mins
Finding the coefficients
On casio fx-991EX
On casio fx-CG50
n
For both nCr and x!
OPTN &gt; PROB
Cr button is shift &divide;
x! is shift x-1
44
P2 – Finite Binomial Expansion where n is a positive integer
This video shows a vertical layout of the Binomial.
Vertical layout of binomial expansion
45
11 mins
P2 - Infinite Binomial Expansion where n is fractional
or negative
Extend to any rational n, including its use for approximation; be aware that the
𝑏𝑥
expansion is valid for | | &lt; 1 (proof not required)
𝑎
May be used with partial fractions. Include the range of validity.
https://youtu.be/JWSBHTjy2os
17 mins
In formula book
Next two terms
1. Expand (1 – 3x) −2
Why are the sequences infinite?
Validity
What does this mean about the validity of the sequence?
46
2
2. Expand (1 – x) 3 up to the term in x3.
State the values for which the series is valid.
3
3. Expand (4 + x) 2 up to the term in x3.
State the values for which the series is valid.
4. Given √
1+𝑥
1 −3𝑥
2
= A + Bx + Cx … Find A, B and C
47
P2 - Binomial Expansion using Partial Fractions
May be used with the expansion of rational functions by decomposition
into partial fractions. May be asked to comment on the range of validity.
https://youtu.be/wbZQW3l3G0w
Express
5𝑥−1
(2−𝑥)(1+𝑥)
12 mins
in ascending powers of x up to the term in 𝑥 2 and state the range of values
for which it is valid.
48
Prior Knowledge – Horizontal Moments
To prepare for this week’s topic you need a good understanding of forces
and connected particles. If you are stuck refer to 10.3 (page 162) and
10.5 (page 169) in the Statistics and Mechanics Year 1 textbook.
Try these questions
1.
1.
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SM9 – Horizontal Moments - Introduction
Introduction to moments
https://youtu.be/izbxpjNg7Q8?t=7
11 mins
What does the moment of a force measure? …………………………………………
How do you calculate the moment of a force? ………………………………………
The anti-clockwise MOMENT about O =
……… X ………
[Nm]
For a body in equilibrium the CLOCKWISE (CW) moments are ………………
to the ANTI-CLOCKWISE (ACW) moments.
Example 1
M(O) :
Resultant Moment = …………… x …………….
= …………………… [Nm]
If the ruler is now in EQUILIBRIUM then:
THE SUM OF THE MOMENTS (IN EITHER DIRECTION) = ZERO
50
Example 2
If we assume the beam is light (no mass), rigid and in EQUILIBRIUM take
moments about O to find the force F.
M(O):
20 ( ………. ) – F ( ………... )
………….
–
…………. F
F
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= 0
= 0
=
[N]
SM9 – Horizontal Moments - Beams
Rigid bodies in equilibrium
https://youtu.be/DQqF8PpMsHs?t=8
20 mins
The most important step in solving MOMENT problems is FIRST to
draw a FORCE DIAGRAM.
PLEASE NOTE THAT IF THE BEAM IS NOT UNIFORMLY DISTRIBUTED WE
CANNOT ASSUME THAT ITS WEIGHT ACTS THROUGH THE CENTRE!!
• Draw the forces acting on the beam above - take
moments about C - equate forces vertically - take