Download 1 Topic 1 Foundation Engineering A

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Topic 1
Vectors and Scalars
Contents
1.1
1.2
1.3
1.4
1.5
1.6
Introduction . . . . . . . . . . . . . . . . . . .
Vector & Scalar Quantities . . . . . . . . . . .
Resultant of Two Vectors . . . . . . . . . . . .
Resultant of Several Vectors . . . . . . . . . .
Three Forces in Static Equilibrium . . . . . .
Resolving a Vector into Components . . . . .
1.6.1 Resolving Vectors and Inclined Planes
1.7 Tutorial Topic 1 . . . . . . . . . . . . . . . . .
1.8 Bibliography . . . . . . . . . . . . . . . . . . .
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Prerequisite knowledge
• Fundamental ability to manipulate algebraic expressions.
• Fundamental understanding of trigonometry.
• Some background in elementary Physics.
• Some familiarity with SI system of units.
Learning Objectives
By the end of this topic, you should be able to:
• Distinguish between vector and scalar quantities.
• Identify fundamental SI Units.
• Use Parallelogram Law & Triangle Law to find resultant of two vectors.
• Find the Resultant of several vectors.
• Resolve vectors horizontally and vertically.
• Solve elementary practical problems posed in terms of vector quantities.
• Solve increasingly difficult problems posed in terms of vector quantities.
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TOPIC 1. VECTORS AND SCALARS
1.1
Introduction
Distinguish between "vector" and "scalar" quantities: a scalar quantity has "magnitude"
(size), but does not have direction; 1 m3 water added to 5 m3 water gives 6 m3 water;
thus, volume can be quantified but has no direction.
1 m3
5 m3
=
6 m3
Vectors, on the other hand, have both magnitude and direction. If an aircraft carrier
is sailing through the Suez Canal with velocity v1 = 5 m s-1 and an aircraft is moving
diagonally across the deck with velocity v2 = 2 m s-1 :
Whenever the overall velocity of the aircraft is needed it is clear that velocities v1 and v2
cannot simply be added - that is 7 m s-1 is not the correct answer:
v2 = 2 m s
-1
v1 = 5 m s-1
- The vectors to the left are
represented by arrows.
- The length of the arrow is the
magnitude.
- The angle it makes to the
horizontal is the direction.
The direction of both the carrier and the aircraft are just as important as the velocities
themselves.
Because vectors (such as velocity) have both magnitude and direction they cannot be
added and subtracted in the same way as scalar quantities (such as volume).
©H ERIOT-WATT U NIVERSITY
TOPIC 1. VECTORS AND SCALARS
1.2
3
Vector & Scalar Quantities
All the fundamental SI units are scalar quantities - they possess magnitude but no
direction. These scalar quantities, their dimensions and associated SI units are listed
below:
Quantity
mass
Dimension
[M]
SI Unit
kg (kilogram)
length
[L]
m (metre)
time
temperature
[T]
s (second)
[θ]
K (kelvin)
amount of substance
[N]
mol (mole)
electric current
luminous intensity
[I]
A (ampere)
[J]
cd (candela)
Other scalar quantities have derived units and sometimes have special names. These
scalar quantities, their names (if any), symbols and SI units are given below:
More Scalar
Quantities
area
Fundamental SI
units
SI Name
SI Symbol
-
-
-
-
m2
m3
kg m-3
pressure
pascal
Pa (N m-2 )
kg m-1 s-2
energy/work/heat
joule
J (N m)
kg m2 s-2
power
watt
W (J s-1 )
kg m2 s-3
frequency
hertz
Hz
angle
radian
rad
s-1
-
volume
density
All the above quantities have only magnitude and cannot be directed in space.
The quantities listed below are all vector quantities - they have both magnitude and
direction. Force, for instance, has magnitude (size) but must always act in a specific
direction - think of friction which always opposes the applied force:
Vector Quantities
displacement
Fundamental SI Units
m
velocity
force
m s-1
m s-2
kg m s-2
momentum
kg m s-1
acceleration
©H ERIOT-WATT U NIVERSITY
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TOPIC 1. VECTORS AND SCALARS
Differentiate between "displacement" and "length". For example, a ruler is 1 m in length
and putting three rulers together gives a total length of 3 m - thus, length has magnitude
only and is a scalar quantity.
However, if a horse gallops 4 km west then 4 km south, what is the overall
√ "displacement"
of the horse relative to the starting point? The answer is of course 32km south-west however, the "distance" travelled is only 8 km.
Thus, distance or length is a scalar quantity and distances can be added in a
straightforward manner. Displacement has both magnitude and direction and is a vector
quantity; the overall displacement (resultant) cannot be found by simple arithmetic.
Velocity, acceleration, momentum and force are all directed quantities and are vectors.
Mathematically there are strict rules governing vector notation and operations - this will
be tackled later, in a rigorous manner, by mathematicians.
In order to assist students at this stage a simplified notation will be used - the
simplification comes in the way the directed aspect of a vector is handled.
For more details and additional worked examples consult standard physics textbooks
[Cutnell and Johnson 2008]
©H ERIOT-WATT U NIVERSITY
TOPIC 1. VECTORS AND SCALARS
1.3
5
Resultant of Two Vectors
Remember the aircraft carrier with velocity v1 = 5 m s-1 sailing through the Suez Canal
and an aircraft, on deck, moving diagonally with velocity v2 = 2 m s-1 - find the resultant
(overall velocity) of these two vectors.
If the two velocity vectors, v1 = 5 m s-1 and v2 = 2 m s-1 , are drawn head-to-tail as shown
below (thin lines) then the resultant (the vector sum) is given by the bold arrow (thick
line) - this is called the Triangle Law of vector summation:
v2 = 2 m s-1
v1 = 5 m s-1
If two force vectors, f1 = 6 N and f2 = 3 N, are drawn tail-to-tail as shown below (thin
lines), first construct a parallelogram (thin lines); the resultant is the bold diagonal arrow
(thick line) - this is called the Parallelogram Law of vector summation:
f2 = 3 N
f1 = 6 N
Either method may be used to find the resultant (vector sum).
Notice unlike scalar quantities, the resultant force or velocity cannot be found using
simple arithmetic - they must be added graphically; however, with a more rigorous
notation (see later Mathematics course) they may be added algebraically.
©H ERIOT-WATT U NIVERSITY
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TOPIC 1. VECTORS AND SCALARS
Example : 1.3.1
Problem:
In an airport a moving walkway operates at 1.2 m s-1 calculate the resultant velocity of
a person who step onto the walkway and does the following:
a) Walks in same direction as the walkway at 1.6 m s-1 .
b) Runs in the same direction as the walkway at 3.0 m s-1
c) Walks in the opposite direction as the walkway at 1.6 m s-1 .
d) Runs in the opposite direction as the walkway at 3.0 m s-1 .
Solution:
It is easy to add vectors when they are in the same, or opposing directions (colinear) - v1
is vector representing walkway, v2 the vector representing person, while r is the resultant
vector (the vector sum):
(a)
v1 = 1.2 m s-1
v2 = 1.6 m s-1
r = 2.8
2 8 m s-11
(b)
v1 = 1.2 m s-1
v2 = 3.0 m s-1
r = 4.2 m s-11
(c)
v2 = 1.6 m s-1
r = 0.4
0 4 m s-1
v1 = 1.2 m s-1
v2 = 3.0 m s-1
(d)
r = 1.8 m s-1
v1 = 1.2 m s-1
Vectors are usually represented by bold lowercase symbols. In the above, the tail of v2
is joined to the head of v1 - the resultant r joins the tail of v1 to the head of v2 . The
resultant (vector sum) is being found using the Triangle Law.
Sometimes it is more convenient to join vectors tail-to-tail, i.e. forces on a body and then
the resultant (vector sum) must be found using the Parallelogram Law.
..........................................
©H ERIOT-WATT U NIVERSITY
TOPIC 1. VECTORS AND SCALARS
Example : 1.3.2
Problem:
A cross country runner runs 3 km east and then jogs 4 km north determine the following:
a) Plot the resultant displacement (the vector sum) on a vector diagram.
b) Find the magnitude of the resultant displacement
c) Find the angle of the resultant displacement with respect to the horizontal axis.
Solution:
Take S1 to be the first displacement and S2 to be the second - the resultant displacement
Sr on a vector diagram is shown below:
a)
sr
s2 =4 km
s1 = 3 km
b) The magnitude of the resultant "Sr " can be found from Pythagoras’ Theorem:
|sr | =
√
32 + 42 = 5 km
c) The direction of the resultant can be found by applying basic trigonometry to the
vector diagram:
∠ Sr = θ = tan−1 4/3 = 53.10
Hence the displacement of the runner, from the original start point, is (5 km, east 53.10◦
north) - notice that the actual distance run is 7 km.
To re-iterate (repeat), distance run is a clearly a scalar quantity, and has magnitude only,
while displacement must be a vector quantity, since it possesses (has) both magnitude
and direction.
..........................................
©H ERIOT-WATT U NIVERSITY
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TOPIC 1. VECTORS AND SCALARS
1.4
Resultant of Several Vectors
Example : 1.4
Problem:
Three forces f1 , f2 and f3 act on a body as shown below. Find the resultant of the three
forces (forces tend to act on a point or a body, so it more convenient to join all the force
vectors tail-to-tail as shown below).
Solution:
When two forces are at right angles, the first step should be to find the resultant of these
two vectors. So taking f1 and f2 and, noting they are joined tail-to-tail, find the resultant
fr1 using the Parallelogram Law - see below:
a)
f2 = 5 N
f2 = 5 N
900
2250
fr1 = 7.1
450
0
225
f1 = 5 N
f3 = 10 N
f1 = 5 N
f3 = 10 N
b) The magnitude and direction of fr1 can be found from Pythagoras’ Theorem:
| fr1 | =
√
52 + 52 = 7.1 N
∠ fr1 = tan−1 5/5 = 450
c) Now fr1 and f3 both lie on same straight line, so overall resultant fr2 is found as
before. The magnitude of fr2 is 2.9 N the angle is 225◦ (in the direction of f3 ):
fr1 = 7.1 N
2250
f3 = 10 N
fr2 = 2.9 N
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©H ERIOT-WATT U NIVERSITY
TOPIC 1. VECTORS AND SCALARS
1.5
9
Three Forces in Static Equilibrium
Example : 1.5
Problem:
Three forces f1 , f2 and f3 act on a body as shown below.
static equilibrium. Find the resultant of these three forces.
The body is in
Solution:
In the previous example, the overall resultant fr2 was shown (thick line) to have
magnitude 2.9 N with an angle to the horizontal of 225◦ (that is in the same direction as
f3 ) - notice that in this case the overall resultant force is not zero.
Such a resultant force would cause this body to accelerate in the direction of f3 - the
applied force causing this acceleration is 2.9 N.
Now consider the point "P" acted on by three forces. However, point "P" is assumed to
be in static equilibrium (simply meaning that point "P" is stationary):
f2
f1
P
f3
Static equilibrium implies (indicates) no unbalanced force then, in this situation, the
overall resultant vector must be zero. Hence, the resultant of f1 and f2 - say fr1 (not
show) - must be equal in magnitude to f3 and opposite in direction.
Therefore, the overall resultant of all three forces fr2 must be zero. The point "P" (or
body) cannot be acted on by any unbalanced force.
This is true for any three forces acting on a body in static equilibrium.
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©H ERIOT-WATT U NIVERSITY
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TOPIC 1. VECTORS AND SCALARS
1.6
Resolving a Vector into Components
Resolving a vector into components is the opposite of finding the resultant of two
vectors. For instance, it is now easy to understand that r is the resultant of vectors
v1 and v2 :
r
v2
v1
But equally, working in the opposite sense, v1 and v2 can be thought of as being
components of r. There are in fact many possible components-pairs of r. However,
in reality, vectors are usually resolved into horizontal and vertical components.
Example : 1.6.1
Problem:
A mail delivery lady pulls her mail trolley with a force of 250 N, while the handle of the
tow bar makes an angle of 30◦ to the horizontal. Calculate the following:
a) The mass needed to keep the trolley from lifting off the ground.
b) The frictional force within the wheels which would prevent any forward motion.
Solution:
The first step is to resolve this vector horizontally and vertically as follows:
- Using trigonometric relations:
………..(A)
f
fy
………….(B)
fx
a) The force acting up is 125 N. The mass needed to keep the trolley from lifting off
the ground = 125/9.81 = 12.7 kg (mail plus trolley).
b) Opposing frictional force which prevents any forward motion = 216.5 N.
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©H ERIOT-WATT U NIVERSITY
TOPIC 1. VECTORS AND SCALARS
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1.6.1 Resolving Vectors and Inclined Planes
Often vectors are resolved horizontal and vertical to some plane along which the vector
is acting. Say a mass "m" is stationary on an inclined plane, then the weight of the body
f (N) is acting vertically while the plane is inclined at an angle "θ":
Inclined Plane
fN
Clearly f may be resolved parallel and perpendicular to the plane (thin lines with arrows).
Notice that all the lines with arrows - the original vector (thick line) and the resolved
vectors (thin lines) - all join tail-to-tail.
To double-check these are indeed components, sketch in a parallelogram as discussed
previously under "Parallelogram Law":
• The thin lines without arrows are the parallelogram construction lines.
• It is obvious from this construction that the resultant of these two component
vectors - using Parallelogram Law - returns the original force vector f.
It should be quite clear that resolving a vector into two components is the exact opposite
of finding the resultant of two vectors.[Cackett Lowrie and Steven 1992]
©H ERIOT-WATT U NIVERSITY
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TOPIC 1. VECTORS AND SCALARS
Example : 1.6.2
Problem:
A body weighs 40 N in the earth’s gravitational field, but rests on a plane inclined at an
angle θ (20◦ ) to the horizontal. If the body is stationary, that is in "static equilibrium",
determine the following:
a) Resolve the weight f vector (weight is a force different from mass) into a
perpendicular vector component "fv " and a parallel vector component "fh " (that
is perpendicular and parallel to the plane).
b) Using data above calculate numerical values for these two component vectors.
c) Calculate the reaction exerted by the plane "frxn " on the mass and the frictional
force ff exerted on the block by the rough surface of the plane.
Solution:
a) As before draw f, the weight vector of the body acting vertically in the earth’s
gravitational field (thick line with arrow) and then resolve this perpendicular (at
right angles) and parallel to the plane - see thin lines with arrows - as follows:
- To solve this problem it should be
recognised that the second angle
beside f is also .
frxn
ff
Inclined
fh
- Use trigonometry to find fv and fh
Plane
fv
fv = f cos
……………………….(A)
fh = f sin
……………………..(B)
f
The Parallelogram construction lines (thin lines without arrows) show that f is the
resultant of fv and fh . To keep the body stationary ff (friction) must be equal &
opposite to fh while frxn (reaction) must be equal & opposite to fv :
b) fv = f cos θ
fv = 40 cos 20 = 37.6 N
fh = f sin θ
fh = 40 sin 20 = 13.7 N
c) frxn = 37.6 N (opposite to fv )
ff = 13.7 N (opposite to fh )
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©H ERIOT-WATT U NIVERSITY
TOPIC 1. VECTORS AND SCALARS
1.7
13
Tutorial Topic 1
1. One displacement vector sa has a magnitude 3.56 km and points due North. A
second displacement vector sb has a magnitude 5.67 km and points due North:
a) Find the magnitude and direction of sa + sb .
b) Find the magnitude and direction of sa - sb .
c) Find the magnitude and direction of sb - sa .
2. At the moment of take-off, a Boeing 747 would typically have a velocity of 80 ms-1
with an angle of 10◦ to the horizontal runway:
a) Calculate the vertical component of the aircraft’s velocity at the moment of
take-off.
b) Calculate the horizontal component of the aircraft’s velocity at the moment of
take-off.
3. A van is being pulled by two forces that are applied by the two ropes shown in the
diagram. Each force has a magnitude of 4000 N and each is directed at an angle
of 30◦ to the horizontal. Use the Parallelogram Law to calculate:
a) The force required by a single rope that would produce the same overall effect
as the two forces below.
b) The direction in which this single force should act.
Van
©H ERIOT-WATT U NIVERSITY
Two ropes each
pulling with a
force of 4,000 N.
14
TOPIC 1. VECTORS AND SCALARS
4. This is a pure Trigonometry question. The diagram shows sodium and chlorine
atoms in a cubic crystalline lattice. Each edge of the cube is 2.8 x 10-10 m in
length.
a) Determine the diagonal distance from one bottom corner of the cube to the
far top corner of the cube - see below.
b) Determine the angle θ that this diagonal makes with the base of the cube.
5. Alex and Jane meet at a street corner and have a little chat. They then go their
separate ways. Alex walks due East along a level street, while Jane goes due
North up a steep hill.
When Alex has reached his next street corner, 200m from the first, Jane has
also reached her next street corner, which is up the hill (Jane’s corner is 150m
as measured along the street and 30 m vertically higher than the original street
corner).
Given the above calculate the total displacement between Alex and Jane.
6. A semi-submersible rig is being manoeuvred by three tug boats and is moving
at a constant velocity. Tugs A and B exert a pull of 30 kN each as shown in the
diagram. Tug C is at the stern of the rig and acts in opposition to the pull from A
and B. Calculate the magnitude of the force exerted by C.
A
C
Semi-sub
20o
B
©H ERIOT-WATT U NIVERSITY
TOPIC 1. VECTORS AND SCALARS
15
7. Consider the following "three force problems". Three forces act on a single point.
Determine whether the forces are in equilibrium or not:
3N
90o
4N
127o
143o
5N
8. A car weighs 13,500 N is moving up an incline at constant velocity. The incline
makes an angle of 25◦ with the horizontal. Determine the reaction force (frxn )
acting on the car.
9. A wire is stretched between the tops of two identical buildings. When a tightrope
walker is at the centre of the wire, the tension in the wire is 2500 N. Each section
of the tight rope makes an angle of 6◦ with respect to the horizontal. Determine
the weight (N) of the walker.
2o
“X” N
©H ERIOT-WATT U NIVERSITY
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TOPIC 1. VECTORS AND SCALARS
10. Two forces act on a body. They act at right angles to one another. One of the
forces has a magnitude of 200 N. The resultant force is of magnitude 300 N:
a) Determine the magnitude of the other force.
b) Calculate the angle between the resultant and the 200 N force.
11. An aircraft has a steady speed of 150 km hr-1 and is travelling due North. A strong
wind blows from the North-West with a speed of 60 km hr-1 . What is the velocity
(magnitude and direction) of the aircraft with respect to the ground?
12. [This question will require more work and insight] Three ropes are tied together at
a single point. Their tensions are 50N, 100 N and 125 N. If the system does not
move calculate all the angles between the ropes.
©H ERIOT-WATT U NIVERSITY
TOPIC 1. VECTORS AND SCALARS
1.8
Bibliography
1. Cutnell, John D. and Johnson, Kenneth W. 2012. Introduction to Physics. 9th ed.
Singapore; Wiley.
2. Cackett, Geoff, Lowrie, Jim and Steven, Alistair. 1999. Higher Still Physics.
Oxford:OUP
©H ERIOT-WATT U NIVERSITY
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GLOSSARY
Glossary
Parallelogram Law
The Parallelogram Law of vector addition allows the resultant to be found in
situations where two vectors are joined tail-to-tail; see “Resultant” in this glossary.
Resolving a Vector
Resolving a vector into components is the opposite operation to finding the
resultant of two vectors; the vector is split into two component parts. In most
cases vectors are resolved either vertically and horizontally in the Earth’s gravity
field, or parallel and perpendicular to some line of action.
Resultant
The resultant of two vectors is the overall effect. The resultant of two vectors not
only takes into account the size of the individual component vectors, but also takes
into account their respective directions; the term resultant applies whenever two
vectors are added or subtracted in order to find the overall effect.
Scalar
A scalar quantity has only magnitude. A scalar quantity has no direction. Work,
energy, power, area, volume, density, etc. are all scalar quantities. Because
scalar quantities have no direction they may be added and subtracted using the
normal rules of arithmetic. Velocity is a vector but speed is a scalar, displacement
is a vector but distance (or length) is a scalar. When motion is restricted to a
straight line, always in the same direction, then and only then can velocity/speed
and displacement/distance be used interchangeably; this is because, under these
strict circumstances, vector addition simplifies back to normal rules of arithmetic.
Static Equilibrium
Whenever the net force acting on a body is zero, and the body is not moving, then
it is said to be in static equilibrium. It implies that the body is absolutely stationary
with no tendency to move and that no unbalanced force is acting upon the body.
Triangle Law
The Triangle Law of vector addition allows the resultant to be found in situations
where two vectors are joined head-to-tail; see “Resultant” in this glossary.
Vector
A vector is a physical quantity that has both a magnitude and a direction.
Displacement, velocity, acceleration, force and momentum are all vector
quantities. Addition and subtraction of vectors must be done vectorially; that is,
not using the normal rules of arithmetic. In this course vectors will be added or
subtracted using vector diagrams. Care must be taken, when adding or subtracting
co-linear vectors, to apply the correct sign convention; co-linear vectors act along
the same straight line, but may be act in the same, or the opposite, direction.
©H ERIOT-WATT U NIVERSITY