Addition and resolution of forces

1
Addition and resolution of forces
2
1 Which of the following schoolbags would you
feel heavier to carry if the same number of books
are put in them?
A
B
3
2 A light rope is stretched tightly between two
poles.
A T-shirt of weight 8 N is hung at the midpoint
of the rope.
What is the tension T in the string? A T lt 4
N B T 4 N C T gt 4 N
8 N
4
Adding forces
a Graphical method

• Force is a vector quantity.
• It has both magnitude and direction.

Like displacements, forces can also be added
graphically using the 'tip-to-tail' method.
Lines representing forces should be drawn to
scale.
5
Adding forces
a Graphical method
'Tip-to-tail' method
resultant force
6
Adding forces
a Graphical method
Parallelogram of forces method
parallel
parallel
resultant force
7
Adding forces
a Graphical method
Video
Simulation
8
Addition of forces
Stretch a rubber band with a spring balance.
F
A
O
B
Stretch the rubber band by the same amount using
2 spring balances.
Note the magnitude direction of the F1 F2.
Check if resultant F (in magnitude direction)
by
using 'tip-to-tail' method
using parallelogram of forces method
9
Addition of forces
Video
10
1 Adding forces
b Algebraic method (1-dimension)
The magnitude of the resultant force is the
algebraic sum of the forces.
Adding forces in the same direction along the
same line.
Adding forces in the opposite direction along
the same line.
F
F
direction is the same as the forces
direction is the same as the larger force
11
1 Adding forces
c Algebraic method (2-demensions)
Forces in 2 dimensions can also be added
algebraically. See this example
12
Example 8
C
Use 1 cm to represent 25 000 N
Length of AC 6.9 cm
resultant
Resultant
D
B
6.9 ? 25 000 N
173 000 N
100 000 N
100 000 N
(with an ? of 30? to either force)
60?
A
oil rig
13
Example 9
2 forces, 3 N and 4 N, act at right angles to
each other. Find the magnitude direction of the
resultant force.
R
5 N
tan ?
5 N
? 36.9?
?
36.9?
Resultant force is 5 N with 36.9? to the 4-N
force.
14
Q1 If Janice is pushing with...
If Janice is pushing with a force of 60 N
Tommy is pushing with 80 N,
what are the maximum and the minimum magnitudes
of their combined force?
Maximum force Minimum force A
100 N 80 N B 120 N 60 N C
140 N 20 N D 160 N 40 N
15
Q2 Find the resultant of the...
Find the resultant of the forces below with
graphical method.
(1 cm represents 1.5 N)
The resultant force is ________ N making an angle
__________ with the 3-N force.
8.9
43?
16
Q3 2 ropes are used to pull...
2 ropes are used to pull a tree as shown.
Find the magnitude of the resultant force acting
on the tree.
120 N
Magnitude of resultant ____________
(Pythagoras theorem) ________ N
170
120 N
17
2 Resolving forces into components
We 've learnt how to combine 2 forces into 1.
Let's see how to spilt (resolve) 1 force into 2.
18
Resolving forces into components
a Graphical method
Suppose a force F is represented by OC
y
component
C
Magnitudes of Fx Fy can be measured directly.
F
component
?
x
O
19
Resolving forces into components
b Algebraic method
Magnitudes of Fx Fy can be also be found by
algebraic method
y
C
F 2 Fx2 Fy2
(Pythagoras theorem)
Fx F cos ?
F
Fy F sin ?
Fy
tan ?
?
Fx
O
x
20
Resolving forces into components
b Algebraic method
Simulation
21
Example 8
2 tug boats pull an oil rig, each exerting a
force of 100 000 N, they are at 60? to each other.
Find the resultant force.
22
Example 10
A 1-kg trolley runs down a friction-compensated
runway at a constant speed.
Find the component of weight of the trolley along
runway,
Hence find friction between trolley runway.
mass 1 kg
? 20?
23
Example 10
Component of weight along runway
Weight (W ) mg
10 N
Component of weight along runway
10 sin 20?
3.42 N
24
Example 10
Friction between trolley runway
constant v
friction
component of weight along runway 3.42 N
? 20?
constant speed
?
? net force on trolley 0
? friction 3.42 N (up the runway)
25
Example 11
180 N
180 sin 12? N
12?
180 cos 12? N
Resolve tension along each rope into 2 components
? along line of travel
? normal to line of travel
Components in each direction are added
26
Example 11
Total force along line of travel
180 cos 12?
170 cos 10?
200 cos (10? 15?)
525 N
A
180 N
12?
10?
B
15?
170 N
200 N
C
27
Example 11
Total force normal to line of travel
180 sin 12?
? 170 cos 10?
? 200 cos (10? 15?)
?76.6 N
ve
A
180 N
12?
10?
B
15?
170 N
200 N
C
28
Example 11
525 N
?
R
76.6 N
76.6
tan ?
525
531 N
? 8.30?
Resultant force exerted by the skiers is 531 N at
an angle of 8.30 to the line of travel.
29
Forces in equilibrium
Forces acting on 1-kg mass
weight
2 forces from 2 springs
But it is at rest,
1-kg mass
? net force 0
Implication
Resultant of any 2 forces
in equilibrium
? equal opposite to the 3rd force
30
Q1 A 1-kg block on rest on a...
A 1-kg block rests on a wedge.
Use graphical method to find the weight
components of the block along normal to the
wedge.
O
C
B
30?
A
What is the friction acting on the block?
31
Q1 A 1-kg block on rest on a...
5
Use a scale of 1 cm to represent _____ N.
at rest
O
Length of OB _____ cm
1.75
C
? Weight component along the wedge ______ N
8.75
B
Length of OC _____ cm
1
30?
A
? Weight component normal to the wedge ______ N
5
Since the block is at rest, friction weight
component ____________ (along/normal to) the
wedge ________ N.
along
5
32
Q2 Find the weight components...
Find the weight components of the block by
algebraic method.
at rest
O
C
Weight component along the wedge 10 _______
______ N
B
30?
sin 30?
A
5
Weight component normal to the wedge 10
_______ _______ N
cos 30?
8.66
33
Q3 What happens to the trolleys...
What happens to the trolleys when they are
released ?
smooth pulley
3 kg
2 kg
20?
30?
A The 3-kg trolley moves downwards.
B Both trolleys remain at rest.
C The 2-kg trolley moves downwards.
34
Example 11
Find the magnitude direction of the resultant
force the skiers exert on the speedboat?
180 N
A
12?
10?
170 N
15?
B
200 N
C
35
Example 12
2 spring balances are used to support an 1-kg
mass.
What are the readings of the 2 balances?
F1
F2
30?
1-kg mass
36
Example 12
30?
30?
30?
10 N
The mass is in equilibrium.
? resultant force of F2 weight is equal
opposite to F1
10
10
? F1
20 N
sin 30?
sin 30?
F1
10
10
? F2
17.3 N
tan 30?
tan 30?
F2
37
Example 12
Alternative method by resolution of forces
30?
The mass is in equilibrium.
? net force 0
10 N
horizontal resultant 0
?
vertical resultant 0
horizontal F1 cos 30? F2
? F1 20 N
vertical F1 sin 30? 10
F2 17.3 N