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BTE1013
Tutorial-Resultant of co-planar forces by calculation
Resultant of coplanar forces by calculation
An alternative to the graphical methods of determining the resultant of two coplanar
forces is by calculation. This can be achieved by trigonometry using the cosine
rule and the sine rule, as shown in Problem following, or by resolution of forces
Problem:
Use the cosine and sine rules to determine the magnitude and direction of the
resultant of a force of 8 kN acting at an angle of 50° to the horizontal and a force of 5
kN acting at an angle of −30° to the horizontal.
The space diagram is shown in Figure 3.11(a). A sketch is made of the vector
diagram, oa representing the 8 kN force in magnitude and direction and ab
representing the 5 kN force in magnitude and direction. The resultant is given by
length ob. By the cosine rule,
ab2 = oa2 + ab2 - 2(oa)(ab)cos∠oab
= 82 + 52 - 2(8)(5)cos100o
(since ∠oab = 180o -50o - 30o = 100o)
= 64 + 25 - (-13.892) = 102.892
Hence
ob = √102.892 = 10.14 kN
By the sine rule,
from which,
5
𝑠𝑖𝑛∠𝑎𝑜𝑏
=
sin ∠aob =
10.14
sin 100°
5𝑠𝑖𝑛100°
10.14
= 0.4856
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BTE1013
hence ∠aob = sin-1(0.4856) = 29.05o.
Thus angle 𝜙 in Figure 3.11 (b) is 50o – 29.05o =20.95o
Hence the resultant of the two forces is 10.14 kN acting at an angle of 20.95° to
the horizontal
Exercise
Further problems on the resultant of coplanar forces by calculation
1. Forces of 7.6 kN at 32° and 11.8 kN at 143° act at a point. Use the cosine and
sine rules to calculate the magnitude and direction of their resultant.
[11.52 kN at 105°]
In questions 2 to 5, calculate the resultant of the given forces by using the cosine
and sine rules
2. 13 N at 0° and 25 N at 30°
[36.8 N at 20°]
3. 1.3 kN at 45° and 2.8 kN at −30°
[3.4 kN at −8°]
4. 9 N at 126° and 14 N at 223°
[15.7 N at −172°]
5. 0.7 kN at 147° and 1.3 kN at −71°
[0.86 kN at −101°]
Exercise
Short answer questions on forces acting at a point
1. Give one example of a scalar quantity and one example of a vector quantity
2. Explain the difference between a scalar and a vector quantity
3. What is meant by the centre of gravity of an object?
4. Where is the centre of gravity of a rectangular lamina?
5. What is meant by neutral equilibrium?
6. State the meaning of the term ‘coplanar’
7. What is a concurrent force?
8. State what is meant by a triangle of forces
9. State what is meant by a parallelogram of forces
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BTE1013
10. State what is meant by a polygon of forces
11. When a vector diagram is drawn representing coplanar forces acting at a point,
and there is no resultant, the forces are in . . . . . . . . . .
12. Two forces of 6 N and 9 N act horizontally to the right. The resultant is . . . . . N
acting . . . . . . .
13. A force of 10 N acts at an angle of 50O and another force of 20 N acts at an angle
of 230O. The resultant is a force . . . . . . N acting at an angle of . . . . O
14. What is meant by ‘resolution of forces’?
15. A coplanar force system comprises a 20 kN force acting horizontally to the right,
30 kN at 45O, 20 kN at 180 O and 25 kN at 225 O. The resultant is a force of . . . . .
.
N acting at an angle of . . . . O to the horizontal
Exercise
Multi-choice questions on forces acting at a point
(Answers on page 284)
1. A physical quantity which has direction as well as magnitude is known as a:
(a) force (b) vector (c) scalar (d) weight
2. Which of the following is not a scalar quantity?
(a) velocity (b) potential energy (c) work (d) kinetic energy
3. Which of the following is not a vector quantity?
(a) displacement (b) density (c) velocity (d) acceleration
4. Which of the following statements is false?
(a) Scalar quantities have size or magnitude only
(b) Vector quantities have both magnitude and direction
(c) Mass, length and time are all scalar quantities
(d) Distance, velocity and acceleration are all vector quantities
5. If the centre of gravity of an object which is slightly disturbed is raised and the
object returns to its original position when the disturbing force is removed, the object
is said to be in
(a) neutral equilibrium
(b) stable equilibrium
(c) static equilibrium
(d) unstable equilibrium
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6. Which of the following statements is false?
(a) The centre of gravity of a lamina is at its point of balance.
(b) The centre of gravity of a circular lamina is at its centre.
(c) The centre of gravity of a rectangular lamina is at the point of intersection of its
two sides.
(d) The centre of gravity of a thin uniform rod is halfway along the rod.
7. The magnitude of the resultant of the vectors shown in Figure 3.29 is:
(a) 2 N (b) 12 N (c) 35 N (d) −2 N
8. The magnitude of the resultant of the vectors shown in Figure 3.30 is:
(a) 7 N (b) 5 N (c) 1 N (d) 12 N
9. Which of the following statements is false?
(a) There is always a resultant vector required to close a vector diagram
representing a system of coplanar forces acting at a point, which are not in
equilibrium.
(b) A vector quantity has both magnitude and direction.
(c) A vector diagram representing a system of coplanar forces acting at a point when
in equilibrium does not close.
(d) Concurrent forces are those which act at the same time at the same point.
10. Which of the following statements is false?
(a) The resultant of coplanar forces of 1 N, 2 N and 3 N acting at a point can be 4 N.
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(b) The resultant of forces of 6 N and 3 N acting in the same line of action but
opposite in sense is 3 N.
(c) The resultant of forces of 6 N and 3 N acting in the same sense and having the
same line of action is 9 N.
(d) The resultant of coplanar forces of 4 N at 0°, 3 N at 90° and 8 N at 180° is 15