Analytical Toolbox

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Analytical Toolbox

• Vectors and Applications
• By
• Dr J.P.M. Whitty

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Learning objectives

• After the session you will be able to
• Explain two types of physical quantities
• Create graphical representation of vector
  quantities
• Resolve vectors
• Perform vector addition
• Use math software (or otherwise) to solve systems
  of vectors in order to answer examination type
  questions

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Scalar Quantities

• Definition
• A scalar quantity is described by a single
  alone (i.e. magnitude) examples include
• Length
• Volume
• Mass
• Time

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Vector Quantities

• Definition
• A vector quantity is described by a magnitude AND
  a direction
• Force
• Velocity
• Acceleration
• Displacement

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Vector Quantities Cont

• A vector quantity (such as force) can be depicted
  as an arrow at an angle to the horizontal, e.g.
  10 Newtons acting at 30 degrees

10N
300
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Example

• Which of the following are vector and which are
  scalar quantities
• Temperature at 373K
• An acceleration downwards of 9.8ms-2
• A weight of mass 7kg
• 500
• A north-westerly in of 20 knots

Scalar
Vector
Vector
Scalar
Vector
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Vector Representations

• A (position) vector may join two points in space
  (A and B say), then, we may say

B
a
Bold face
A
They are usually written as
With the magnitude written as
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Equal vectors

• Two vectors are equal if they have the same
  magnitude and direction

B
D
a
c
A
C
Here we say
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Equal and opposite vectors

• Two vectors are equal and opposite then they
  have the same magnitude but act in opposite
  directions (sometimes referred to as negative
  vectors)

B
D
c
a
A
C
Here we say
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Addition of vectors

• Any quantities can be added using the a
  parallelogram of triangular rules

Resultant vector
Parallelogram rule Vectors drawn from a single
point
Triangular rule Vectors placed end to end
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Example 1

• Find the resultant force of two 5N_at_10o and
  8N_at_70o

R
8Units _at_70o
8Units _at_70o
5Units _at_10o
5Units _at_10o
Measure R to give 11.4N_at_42o
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Example 2

• Find the resultant force of three 6N_at_5o and
  8N_at_40o and 10N_at_80o

R
Solution 1 Apply the Parallelogram rule twice
10Units_at_70o
Measure R to give 21.6_at_44o
8Units_at_40o
6Units_at_5o
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Example 2 Triangular rule

• Here we simply place the vectors end to end,
  thus

R
Measure R to give 21.6_at_44o
10Units_at_70o
8Units_at_40o
6Units_at_5o
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Sum of a number of vectors

• In general the triangular rule takes less
  construction and it is also easier extended to
  account for a number of vectors, thus
• Let a be a vector from A to B, b be from B to C
  and so on
• Then abcd, can be evaluated by formation of
  vector chain

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Vector chains

• The sum abcd, is constructed thus

Here we can say
E
d
Notice the pattern
D
r
c
A
C
a
b
B
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Example

• Given that P,Q,R and S are point in three
  dimensional space, find the vector sum of

Solution
This has the same pattern as previously, i.e. a
connected path thus
Note No need to draw the diagram the outside
letters render the result so long as they are
connected
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The null vector

• Suppose we consider another case where the
  resultant vector r-e, we have

Here we now have
E
d
D
i.e. the same position
e
c
A
C
i.e. 0, the null vector, has no length hence
direction
a
b
B
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Class Examples Time

• Find the sum of the position vectors

Solution
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Vector components

• The vector OP is defined by its magnitude r and
  its direction ?. It can also be defined in terms
  of the components a and b in the directions OX
  and OY, respectively.

y
P
r
b
?
x
O
a
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Unit vectors

• Hence OPa (along OX) b (along OY).
• If a unit vectors i,j (i.e. vectors of length
  unity) are introduced along OX and OY
  respectively then
• ra i b j i a j b
• r i rcos? j rsin?
• Where a and b are the lengths along OX and OY,
  equal to the magnitudes of the original vectors

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Vector addition (analytic solution)

• The use of unit vector allows the calculation of
  vector addition analytically. Returning to the
  previous example 2, viz

Find the resultant force of three 6N_at_5o and
8N_at_40o and 10N_at_80o
Here the solution is to resolve the vectors into
components and add them to give the overall result
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Example 2 analytic solution

• Letting the forces equal F1, F2 and F3
  respectively
• F1 i rcos? j rsin? i6cos5o j6sin5o
  5.977i0.526j (3dp)
• F2 ircos? jrsin? i8cos40oj8sin40o
  6.128i5.142j (3dp)
• F3i rcos? j rsin?i10cos70oj10sin70o
  3.420i9.397j (3dp)
• Therefore adding the individual components
• r15.525i15.065j (3dp)

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Example 2 analytic solution

• The result is in Cartesian form, however we have
  been asked for the magnitude and direction of the
  resultant vector. To do this we must resort back
  to elementary trigonometry and Pythagoras, thus

y
P
r
b15.065
?
x
O
a15.525
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Example 2 Alternative notation

• The previous example can be evaluated using
  column or row vectors as follows

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Example 2 MatLab

• This notation allows to solve such problems using
  math software such as MatLab

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Example 3

• Find the forces in the members of the structure
  and evaluate the stress in each given that each
  bar is 50mm in diameter

C
60o
B
A
300N
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Example 3 Solution

• i(FAB)-i(FBCCos60o)j(FBCSin60o)j300

i FAB-FBCCos60o0 j (FBCSin60o)300
j
C
i
60o
B
A
300N
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Example 4

• Find the forces in the members of the structure
  and evaluate the stress in each given that each
  bar is 25mm in diameter

B
60o
500N
A
C
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Example 4 Solution

• Apply vector eqn, thus
• i(FABcos30o) j(FABsin30o)- i(FBCcos30o)
  j(FBCSin30o)j500

B
60o
500N
A
C
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Examination type questions (Homework)

1. Find the value of the resultant force given that
   the following act on a specific point in a roof
   truss.

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Examination type questions (Homework)

• Given that the following three forces act on a
  12mm diameter bar

Find the resultant force on the bar 3, and
evaluate the maximum stress that bar can
experience.
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Examination Type Question

• Exploit symmetry conditions and find the stresses
  in each of red members the 20mm dia, steel
  members (E200GPa). Hence or otherwise evaluate
  the resulting strains.
• 20 marks

1m
1m
250
250
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Examination Type Solution

• Exploit symmetry thus

B
C
A
250
250
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Examination Type Question Strain value solutions

• These can be evaluated from the elasticity
  definitions as well!

Note the units here are of utmost importance
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Examination Type Question Stress value solutions

• These can be evaluated from the elasticity
  definitions

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Summary

• Have we met our learning objectives specifically
  are you now able to
• Explain two types of physical quantities
• Create graphical representation of vector
  quantities
• Resolve vectors
• Perform vector addition
• Use math software (or otherwise) to solve systems
  of vectors in order to answer examination type
  questions