FORCES 3. Calculating resultant forces
Doc Brown's Physics Revision
Notes
Suitable for GCSE/IGCSE Physics/Science courses or
their equivalent
This page will answer questions such as ...
What is a resultant force? Why is it a
vector?
How do we draw scale diagrams to deduce
a resultant force?
What do we mean by balanced and
unbalanced forces?
When dealing with acting forces, what
is an equilibrium?
Introduction
Forces were introduced in "What is a force?"
including contact and noncontact forces AND, importantly for this page, free
body diagrams showing multiple forces on an object.
Force data is useless without its direction
of action. You not only need to know the value of a force in newtons (N) but the
precise direction or angle of the line of action of one force with respect to at
least one other force.
That is why force is always a vector  it
has magnitude and direction!
When a body is subjected to multiple known
forces (usually >= vectors in newtons) how can we deduce and calculate the net
resultant force and its direction?
It is possible to replace multiple forces
acting at a single point with a single force known as the resultant force.
Some resultant forces are easily deduced with
a simple addition or subtraction.
Others require a scale drawing showing all
the forces involved and the direction (e.g. angle) in which each individual
force acts.
From such graph work you can measure and
calculate the resultant force and its direction of action.
Examples of problem
solving to calculate a resultant force
See also
work done
calculations in Appendix 2.
Q1.
Two forces acting in the same direction (parallel) on an object
What is the resultant force on the object?
In this case you simply add up the two forces acting
from left to right:
resultant force = force 1 + force 2 = 70 N + 45 N
= 115 N
So the object experiences an acting force of
115 N to the right (vector!). 
Q2.
Two forces acting in parallel but opposite directions
What is the resultant force on the object?
Above you added the forces, but here, you subtract
one from the other.
resultant force = force 1  force 2 = 780  330 =
450 N to the right
Since the force to the right is greater than the
opposing force from the left, the net force (resultant force of 330 N)
must act from left to right. 
Q3.
Multiple force acting on an object Given
the free body diagram, what is the magnitude and direction of the
resultant force.
This
situation is not quite as complicated as it seems because acting forces 2
and 3 are equal and cancel each other out i.e. they have no effect
on the net resultant force. Whatever the motion of the object, it will
not rise or fall.
Therefore the situation is actually the same as
example Q2. above.
resultant force = force 1  force 4 = 120  55 =
65 N horizontally to the right This
free body diagram could represent a moving vehicle. 
Q4.
Resolving two forces that are not parallel
Imagine an object is subjected to a northward force
of 90 N and a due east force of 60 N.
Deduce the magnitude of the resultant force and
its angle with respect to due north.
Using graph paper and a suitable scale, you draw
the two forces out at 90^{o} to each other in the manner shown in the left
diagram ('tip to tip')  which produces a triangle when you join up the start of the
north force to the right hand tip of the east force.
You then measure the length of the hypotenuse and
from the chosen scale you get 7.2 cm which translates to 7.2 x 15 =
108 N.
Using a protractor you can then measure the angle
which I found to be 33^{o} (033^{o}) with
respect to north. I made both measurements as a student would in class
using 2 mm graph paper.
Using trigonometric calculations (see
APPENDIX 1):
In GCSE exams you will have to solve it by the
graphical method described above, but you can solve it just from a
simple nonscale sketch by trigonometry using the known direction of
the two forces and the angle between them. This is really advanced A level maths using a scientific
calculator, but I decided to check my own 'honest' graph work just using
the forces and directions given and ignoring the graph completely.
tan θ = O / A = 60/90 = 0.667, tan^{1}(0.667)
= 33.7^{o} (so I made a small manual graph error of 0.7^{o})
You can work out the magnitude of the resultant
force using the accurate angle above and either of the sine or cosine
rule equations e.g. if R = resultant force (= H)
sin θ = O / H = sin (33.7^{o}) = 60/R =
0.555, R = 60/0.555 = 108 N
cos θ = A / H = cos (33.7^{o}) = 90/R =
0.832, R = 90/0.832 = 108 N
So my resultant force measurement was spot on!

Q5.
Resolving a forces into two forces at right angles to each other
Imagine a force of 156 N acting at an angle
of 51^{o} from the vertical 'north'.
Q Deduce the component vectors for due north and due
east.
You draw the resultant force line to scale at 39^{o} (90^{o}
 51^{o}) from the horizontal.
Use a convenient triangle e.g. in this case the
hypotenuse of the triangle is 7.8 cm. and the other two sides are 6.0 to
the east and 5.0 cm to the north.
I've then chosen a scale of 20.0
N/cm on 2 mm graph paper.
( Note from the scale chose 7.8 x 20.0 = 156 N)
You can then split this 'resultant' force into two
components by draw lines for the northerly contribution
(5.0 cm) and the easterly contribution (6.0 cm). The convenient numbers
of 5.0 and 6.0 was pure coincidence!
From the scale this gives the two contributing
forces of:
5.0 x 20.0 = 100 N to the north and 6.0
x 20.0 = 120 N to the east.
Again you can work out the answers from
trigonometry just using the angle of 39^{o} and the force of 156
N (the original information): Let N be the north force and E be the east
force
sin θ = O / H = sin (39^{o}) = 0.629 =
N/156, N = 0.629 x 156 = 98.1 N to the north
cos θ = A / H = cos (39^{o}) = 0.777
= E/156, E = 0.777 x 156 = 121.2 N to the east
So I'd made a 12% error on my graph work, and
note the trigonometrical calculations are absolutely correct based on
the original information. This level of calculation is covered by GCSE maths
courses. 
Q6.
Variations on two forces.
Situation A: A pulling force of 225 N is acting on
an object at an angle of 27^{o} to the horizontal. At the same
time it is also subjected to another pulling force at 45^{o} to
the horizontal (as shown in the diagram).
With graph paper, protractor and ruler, draw a scale diagram to deduce the magnitude and
direction of the resultant force. The graph is drawn to a scale of 1 mm
= 5 N.
This can be solved using the principle of 'parallelogram
of forces'. Its quite simple to do. The dotted line are drawn
parallel to the two known vectors and where they intersect gives you the
length of the resultant force of 60 mm.
Therefore the resultant force is 60 x 5 =
300 N horizontally to the right.
However, you can avoid the parallelogram of forces
diagram by drawing the force diagram in a simpler way as shown in
diagram B.
You draw the first force of 225 N at 27^{o}
to the horizontal (45 mm) and from its top tip draw the 2nd force
downwards at 45^{o} to the horizontal line (28 mm).
The horizontal line of 60 mm gives you the same
resultant force of 300 N.
In
situation C I want you to image the 300 N force acting in the opposite
direction. In this case there is NO net
resultant force. The object is then described as being in a state of
equilibrium and would remain stationary.
This illustrates a method of deducing a force needed
to produce an equilibrium situation involving three forces e.g. you
might be given two and have to work out the 3rd force needed to balance
the other two forces. more examples needed 
Q7.
A a hanging weight, tensioned horizontally
A mass of 20 kg is suspended from a beam by a wire at
angle of 32^{o} away from the vertical (see diagram). The
tension in this wire is T2. The same 20 kg weight is pulled to one side
by a horizontal string with a tension T2.
Using a graph calculate the tension force in (a)
the wire (T1) and (b) in the string (T2).
The triangle of forces is drawn on 1 mm graph paper.
Tension T3 is the weight of the object = 196 N (20 x 9.8, g = 9.8 N/kg).
This is the vertical force.
The diagonal side of the triangle is tension T1 in
the wire holding up the weight at an angle of 32^{o}.
The horizontal side of the triangle is the tension
T2 in the string pulling from the left.
(a) The graph is drawn to a scale of 2 N per mm.
The diagonal (hypotenuse) was found to be 115 mm long.
From the scale tension T1 = 115 x 2 =
230 N
(b) The opposite side of the triangle = 60 mm,
this equates to 60 x 2 = 120 N for tension T2 in the string
Trigonometric calculation check (perfect answers!):
Tension T1: cos (32^{o}) = adjacent /
hypotenuse = 0.848 = T3/T1 = 196/T1, so T1 = 196/0.848 = 231.1
N Tension T2: tan (32^{o})
= opposite / adjacent = 0.625 = T2/T3 = T2/196, so T2 = 0.625 x 196
= 122.5 N So, I think my
graph work was pretty good with only a 0.4% to 0.5% error!

Q8.
Resolving two converging forces Two
forces, 5.0 N and 6.0 N, act on an object at an angle of 60^{o}
between the lines of action (as in the diagram).
Calculate the resultant force on the object O at
point o.
Using the principle of the parallelogram of
forces, and focus on the left of the diagram: If you draw the lines ab
(6.0 cm) parallel to action line oc and line bc (5.0 cm) parallel to
action line ao, then the diagonal of 9.5 cm gives you the resultant
distance which equates to a resultant force of 9.5 N (1 cm
= 1 N).
However, drawing true parallel lines is awkward,
so you can deduce the answer without graph paper and on plain white
paper with just a mm ruler and protractor. If you extend the line ao a
distance of 5 cm giving line od, you get exactly the same resultant
distance of 9.5 cm = resultant force of 9.5 N by joining up the relevant
tip to tip points cd.
Its simple pure geometric logic, no problem! Just
compare situations A and B in Q6. 
Q9.
Calculating a 3rd force required to establish an equilibrium
Two wires with pulling force tensions of 20 N and 24
N are pulling on a metal ring. If the angle between the lines of action
is 70^{o} calculate the force needed on the 3rd wire on the
right to stabilise the ring to give an equilibrium situation.
Using the principle of the parallelogram of
forces, and focus on the left of the diagram: If you draw the lines ab
(6.0 cm) parallel to action line oc and line bc (5.0 cm) parallel to
action line ao, then the diagonal of 9.0 cm gives you the resultant
distance. BUT, the direction of action is opposite to that in Q8, so
this equates to a resultant force of 36.0 N (1 cm = 14N)
and the arrow points towards the right balancing and opposing the forces
acting to the left.
As in Q8, drawing true parallel lines is awkward,
so you can deduce the answer without graph paper and on plain white
paper with just a mm ruler and protractor. If you extend the line ao a
distance of 5 cm giving line od, you get exactly the same resultant
distance of 9.0 cm = resultant force of 9.0 x 4 =36 N by joining
up the relevant tip to tip points cd.
Note that although the total force to the left is
44 N, the opposing force is less (36 N), because the forces are not
acting parallel to each other. 
Q10.
A push and pull grass roller! A roller
of mass 80 kg is pushed and pulled with a force of 300 N acting at an
angle of 45^{o} to the horizontal grass surface.
Calculate the vertical force of the roller on the
grass when it is (a) pushed and (b) pulled.
You first need to calculate the vertical force
involved due to the person pushing or pulling the roller (call it Fp).
You can do this simply with a graph, but I'll
leave you to practice that and I'll go straight for a trigonometric
calculation.
sin (45^{o}) = opposite/hypotenuse = 0.708
= Fp/300, so Fp = 0.708 x 300 = 212 N (3 sf)
You also need the weight of the roller = m x g =
80 x 10 = 800 N (taking gravity as 10 N/kg)
(a) When pushing the roller you are increasing the
overall downward force of the roller, therefore
total vertical force exerted by the roller =
weight of roller + vertical pushing force
= 800 + 212 = 1012 N, the
vertical normal contact force due to the roller.
(b) When pulling the roller you are decreasing to
overall downward of the roller, therefore
total vertical force exerted by the roller =
weight of roller  vertical pulling force
= 800  212 = 588 N, the
vertical normal contact force due to the roller.
AND, now you can see why its easier to pull a
roller than push it and how accurate was your graph work! 
Q11.
A suspended microphone! A microphone of
mass 750 g is suspended by a wire at 25^{o} to the vertical
(tension T2, diagram) and pulled to the right by a horizontal cord
(tension T1, diagram).
(a) Calculate the tension in the wire holding up
the microphone.
(b) What tension must be applied to the horizontal
cord to maintain the microphone in a stable position?
You can do this simply with a graph as in Q7, but
I'll leave you to practice that and I'll go straight for trigonometric
calculations, also at the end of Q7 and as in Q11. above.
750 g = 0.75 g, so m = m x g = 0.75 x 10 = 7.5
N for the weight of the microphone.
The diagram shows how the forces will operate.
(a) The tension in the wire T2
cos (25^{o}) = adjacent/hypotenuse =
0.9063 = 7.5/T2, so T2 = 7.5/0.9063 = wire tension = 8.27 N
(3 sf, 2 dp)
Note that the tension on the wire is greater
than the weight of the microphone because it is being pulled both
downwards and to one side.
(b) Force T1 needed to stabilise microphone
tan 25^{o}) = opposite/adjacent =
0.466 = T1/7.5, so T1 = 0.466 x 7.5 = cord tension required =
3.50 N (3 sf)
How accurate was your graph work! 
Q12 The motor
of a ferry boat provides driving force of 250 N perpendicular to the
bank of a river. If the flow of the river acts in the same direction
as the river bank with a force of 100 N ...
(a) Calculate the magnitude and direction of the resultant force.
By drawing a graph diagram or from a trigonometric calculation
you should get a resultant force of
269 N.
Ignoring the rudder! the boat will travel away from the bank at
an angle of
68^{o}
to it (or 22^{o} from the perpendicular line from the bank).
(b) With reference to your answer to (b), how should the ferry be
steered to minimise the distance travelled to cross the river?
To counteract the flow of the river, the boat should be steered
at an angle opposite to the direction indicated above in (a). This
means working against the flow of the river to counteract the
enforced sideways movement of the boat.
You don't save energy but you can reduce the distance travelled.

Q13 
APPENDIX 1:
right angle triangle rules
Important formulae in the trigonometry of a
right angled triangle:
for angle θ shown on the right diagram
tangent rule: tan θ = ^{opposite} / _{adjacent}
sine rule: sin θ = ^{opposite} / _{hypotenuse}
cosine rule: cos θ = ^{adjacent} / _{hypotenuse}
APPENDIX
2 Calculating work done from a resultant force
If a source of energy is available, you can
calculate the work done from the acting force and the distance the force acts
through.
work done (joules) = force (newtons) x
distance along the line of action of the force (metres)
W (J) = F (N) x d (m), F =
W/d, d = W/F
e.g.
resulting
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