,UNIT 1 - VECTORS
Vectors and Scalars
Any quantity that had magnitude only is called a scalar quantity. If a quantity has direction and magnitude then it
is a vector quantity. We use arrows to represent vector quantities.
Tail of Arrow - indicates & Head of Arrow —> direction
starting point it is acting
Length of Arrow —> magnitude
or size of vector
Vector Diagrams —> always declare your choice of the positive direction. These shows the direction and magnitude
of force
The Resultant —> when more than one vector acts in a system, a net or overall effect needs to be determined by
adding or subtracting them. This is called a resultant vector and it can only be calculated with vectors of the same
quantities ( only velocity or only forces ).
The Equilibrant —> a vector which has the same magnitude as a resultant but acts in the opposite direction
Vectors in 1D
Resultant of Vectors in 2D
If vectors are in different directions, we use geometry to determine magnitude of the sum and the direction
a
Fur
eg. Force of 20N acts Magnitude Pythagoras Direction a
↓.
trigonometry
e
2)
up and force of 12N FR = (F1)2 +
(Fz)7 rang =
y/x
acts to the right. FR = 202 + 12 +ano =
12/20
Calculate resultant I
0
23 ,
32No 960
Fr :
& = =
30 ,
force.
,The Triangle Rule for Forces in Equilibrium
A system is in equilibrium when a number of forces which are acting on
↑
j Fl
the same object cancel the effect of each other out. These forces keep &
F2
the object in equilibriums so that the object does not move. If it does F3
move, it does so with constant velocity. When 3 forces which act on an L
&
object in equilibrium, a closed vector diagram or triangle will be F2
obtained. The resultant force is 0 therefore will not appear in the j F3
diagram. 3rd force is equal in magnitude but opposite in direction of the Fl
V
resultant of the other 2 forces. These forces are balanced.
Components of Vectors
When we resolve components, we resolve them into vertical ( y-axis ) and horizontal ( x-axis ) components. They
will be perpendicular to each other hence trig can be used. Cos is used for horizontal components and sin is
used for vertical components. Remember that if the angle of bearing it on large, work with the smaller angle in
the triangle ( both shown in examples ).
eg. Force of 500N act at an angle of 40° to the horizontal
Fig
28
N Horizontal : vertical :
5
DON Fx
Fx =
F .
COSO Fy
=
F .
Sin &
M LOS 400 500
:
Sin 40
# 2 500 X
Fy
= =
.
480
Fx =
383 , 02N Fy
=
321 39N
g) ,
g
eg. Force of 5,5N acts on bearing of 210°
ga
0 =
2700 -
2189 Horizontal : vertical :
int
=
600 Fx F COS & F sin O
Fy
= =
. .
:
Fx 5,5 x COS 60 ,
5 5 Sin 60a
Fy
= = .
Fx 2 ,
75N Fy 4 , 76N
=
=
,
H
L
UNIT 2 - MOTION CONCEPTS
Positions and Frame of Reference —> the position of an object describes the object’s location and it is always
described relative to a known reference point. Frame of reference is a reference point with a set of directions
associate with it. It must have a point of origin in and any direction away from that origin must be assigned a
positive or negative value.
Distance and Displacement —> distance is a scaler quantity that is defined as the length of the path travelled
and displacement is a vector quantity and is defined as a change in position
, Speed and Velocity —> speed is a scalar quantity and is defined as the rate of change of distance and it can only
be positive. Velocity is a vector quantity and is defined as the rate of change of position or rate of change of
displacement and it can be positive or negative. Instantaneous speed is at any point in time and average speed is
for the entire journey and this is when the total distance travelled is used as well as total time. This is similar to
average velocity and instantaneous velocity except displacement is used.
Speed Velocity ( instantaneous )
Speed ( m.s-¹ ) = distance ( m ) s=d Velocity ( m.s-¹ ) = change in displacement ( m ) V= x
time ( s ) t change in time ( s ) t
Average Velocity
V ave ( m.s-¹ ) = total displacement ( m )
total time ( s )
Acceleration —> the rate of change of velocity a ( m.s-² ) = vf - vi ( m.s-¹ )
t(s)
an
W During accelerated motion,
change in velocity is always
constant so there will
always be a 0 gradient.
Vectors and Scalars
Any quantity that had magnitude only is called a scalar quantity. If a quantity has direction and magnitude then it
is a vector quantity. We use arrows to represent vector quantities.
Tail of Arrow - indicates & Head of Arrow —> direction
starting point it is acting
Length of Arrow —> magnitude
or size of vector
Vector Diagrams —> always declare your choice of the positive direction. These shows the direction and magnitude
of force
The Resultant —> when more than one vector acts in a system, a net or overall effect needs to be determined by
adding or subtracting them. This is called a resultant vector and it can only be calculated with vectors of the same
quantities ( only velocity or only forces ).
The Equilibrant —> a vector which has the same magnitude as a resultant but acts in the opposite direction
Vectors in 1D
Resultant of Vectors in 2D
If vectors are in different directions, we use geometry to determine magnitude of the sum and the direction
a
Fur
eg. Force of 20N acts Magnitude Pythagoras Direction a
↓.
trigonometry
e
2)
up and force of 12N FR = (F1)2 +
(Fz)7 rang =
y/x
acts to the right. FR = 202 + 12 +ano =
12/20
Calculate resultant I
0
23 ,
32No 960
Fr :
& = =
30 ,
force.
,The Triangle Rule for Forces in Equilibrium
A system is in equilibrium when a number of forces which are acting on
↑
j Fl
the same object cancel the effect of each other out. These forces keep &
F2
the object in equilibriums so that the object does not move. If it does F3
move, it does so with constant velocity. When 3 forces which act on an L
&
object in equilibrium, a closed vector diagram or triangle will be F2
obtained. The resultant force is 0 therefore will not appear in the j F3
diagram. 3rd force is equal in magnitude but opposite in direction of the Fl
V
resultant of the other 2 forces. These forces are balanced.
Components of Vectors
When we resolve components, we resolve them into vertical ( y-axis ) and horizontal ( x-axis ) components. They
will be perpendicular to each other hence trig can be used. Cos is used for horizontal components and sin is
used for vertical components. Remember that if the angle of bearing it on large, work with the smaller angle in
the triangle ( both shown in examples ).
eg. Force of 500N act at an angle of 40° to the horizontal
Fig
28
N Horizontal : vertical :
5
DON Fx
Fx =
F .
COSO Fy
=
F .
Sin &
M LOS 400 500
:
Sin 40
# 2 500 X
Fy
= =
.
480
Fx =
383 , 02N Fy
=
321 39N
g) ,
g
eg. Force of 5,5N acts on bearing of 210°
ga
0 =
2700 -
2189 Horizontal : vertical :
int
=
600 Fx F COS & F sin O
Fy
= =
. .
:
Fx 5,5 x COS 60 ,
5 5 Sin 60a
Fy
= = .
Fx 2 ,
75N Fy 4 , 76N
=
=
,
H
L
UNIT 2 - MOTION CONCEPTS
Positions and Frame of Reference —> the position of an object describes the object’s location and it is always
described relative to a known reference point. Frame of reference is a reference point with a set of directions
associate with it. It must have a point of origin in and any direction away from that origin must be assigned a
positive or negative value.
Distance and Displacement —> distance is a scaler quantity that is defined as the length of the path travelled
and displacement is a vector quantity and is defined as a change in position
, Speed and Velocity —> speed is a scalar quantity and is defined as the rate of change of distance and it can only
be positive. Velocity is a vector quantity and is defined as the rate of change of position or rate of change of
displacement and it can be positive or negative. Instantaneous speed is at any point in time and average speed is
for the entire journey and this is when the total distance travelled is used as well as total time. This is similar to
average velocity and instantaneous velocity except displacement is used.
Speed Velocity ( instantaneous )
Speed ( m.s-¹ ) = distance ( m ) s=d Velocity ( m.s-¹ ) = change in displacement ( m ) V= x
time ( s ) t change in time ( s ) t
Average Velocity
V ave ( m.s-¹ ) = total displacement ( m )
total time ( s )
Acceleration —> the rate of change of velocity a ( m.s-² ) = vf - vi ( m.s-¹ )
t(s)
an
W During accelerated motion,
change in velocity is always
constant so there will
always be a 0 gradient.
