Transcript Physics and Physical Measurement Topic 1.3 Scalars and Vectors

Physics and Physical
Measurement
Topic 1.3 Scalars and Vectors
Scalars Quantities
 Scalars can be completely described by
magnitude (size)
 Scalars can be added algebraically
 They are expressed as positive or negative
numbers and a unit
 examples include :- mass, electric charge,
distance, speed, energy
Vector Quantities
 Vectors need both a magnitude and a direction to




describe them (also a point of application)
When expressing vectors as a symbol, you need to
adopt a recognized notation
e.g.
They need to be added, subtracted and multiplied in
a special way
Examples :- velocity, weight, acceleration,
displacement, momentum, force
Addition and Subtraction
 The Resultant (Net) is the result vector that
comes from adding or subtracting a number
of vectors
 If vectors have the same or opposite
directions the addition can be done simply
 same direction : add
 opposite direction : subtract
Co-planar vectors
 The addition of co-planar vectors that do not have the
same or opposite direction can be solved by using
scale drawings to get an accurate resultant
 Or if an estimation is required, they can be drawn
roughly
 or by Pythagoras’ theorem and trigonometry
 Vectors can be represented by a straight line
segment with an arrow at the end
Triangle of Vectors
 Two vectors are added by drawing to scale
and with the correct direction the two vectors
with the tail of one at the tip of the other.
 The resultant vector is the third side of the
triangle and the arrow head points in the
direction from the ‘free’ tail to the ‘free’ tip
Example
R=a+b
a
+
b
=
Parallelogram of Vectors
 Place the two vectors tail to tail, to scale and
with the correct directions
 Then complete the parallelogram
 The diagonal starting where the two tails
meet and finishing where the two arrows
meet becomes the resultant vector
Example
R=a+b
a
+
b
=
More than 2
 If there are more than 2 co-planar vectors to
be added, place them all head to tail to form
polygon when the resultant is drawn from the
‘free’ tail to the ‘free’ tip.
 Notice that the order doesn’t matter!
Subtraction of Vectors
 To subtract a vector, you reverse the direction
of that vector to get the negative of it
 Then you simply add that vector
Example
a
-
b
=
R = a + (- b)
-b
Multiplying Scalars
 Scalars are multiplied and divided in the
normal algebraic manner
 Do not forget units!
 5m / 2s = 2.5 ms-1
 2kW x 3h = 6 kWh (kilowatt-hours)
Multiplying Vectors
 A vector multiplied by a scalar gives a vector with the
same direction as the vector and magnitude equal to
the product of the scalar and a vector magnitude
 A vector divided by a scalar gives a vector with same
direction as the vector and magnitude equal to the
vector magnitude divided by the scalar
 You don’t need to be able to multiply a vector by
another vector
Resolving Vectors
 The process of finding the Components of
vectors is called Resolving vectors
 Just as 2 vectors can be added to give a
resultant, a single vector can be split into 2
components or parts
The Rule
 A vector can be split into two perpendicular
components
 These could be the vertical and horizontal
components
Vertical component
Horizontal component
 Or parallel to and perpendicular to an inclined
plane
 These vertical and horizontal components
could be the vertical and horizontal
components of velocity for projectile motion
 Or the forces perpendicular to and along an
inclined plane
Doing the Trigonometry
V
Sin  = opp/hyp = y/V
y

x
V sin 

Therefore y = Vsin 
In this case this is the
vertical component
Cos  = adj/hyp = x/V
Therefore x = Vcos 
In this case this is the
horizontal component
V cos 
Quick Way
 If you resolve through the angle it is
 cos
 If you resolve ‘not’ through the angle it is
 sin
Adding 2 or More Vectors by
Components
 First resolve into components (making sure
that all are in the same 2 directions)
 Then add the components in each of the 2
directions
 Recombine them into a resultant vector
 This will involve using Pythagoras´ theorem
Question
 Three strings are attached to a small metal
ring. 2 of the strings make an angle of 70o
and each is pulled with a force of 7N.
 What force must be applied to the 3rd string
to keep the ring stationary?
Answer
 Draw a diagram

7 cos 35o + 7 cos 35o
7N
7N
70o
7 sin 35o
7 sin 35o
F
 Horizontally
 7 sin 35o - 7 sin 35o = 0
 Vertically
 7 cos 35o + 7 cos 35o = F
 F = 11.5N
 And at what angle?
 145o to one of the strings.