EXAMPLE 9
A 20-kg trunk is resting on a ramp inclined at an angle of 15º. Calculate the
components of the force of gravity on the trunk that are parallel and perpendicular
to the ramp. Describe the physical consequences of each.
Solution
The force of gravity on the trunk is (20 kg) (9.8 m/s2) 196 N acting down.
The parallel and perpendicular components are
Fp
៮៬  196 sin 15º and
F
p
⬵ 51 N
៮៬  196 cos 15º
F
n
⬵ 189 N
15°
196 N
Fn 15°
196 N
position diagram
force diagram
The parallel component points down the slope of the ramp. It tends to cause the
trunk to slide down the slope. It is opposed by the force of friction acting up the
slope. The perpendicular component presses the trunk against the ramp. The
magnitude of the force of friction is proportional to this component.
Exercise 4.3
Part A
Communication
Knowledge/
Understanding
1. Name some common household objects on which the force of gravity is
approximately 2 N; 20 N; 200 N. What is your weight in newtons?
2. Find the horizontal and vertical components of each of the following forces.
a. 200 N acting at an angle of 30º to the horizontal
b. 160 N acting at an angle of 71º to the horizontal
c. 75 N acting at an angle of 51º to the vertical
d. 36 N acting vertically
3. Find the resultant of each pair of forces acting on an object.
a. forces of 7 N east and 12 N west
b. forces of 7 N east and 12 N north
c. forces of 6 N southwest and 8 N northwest
d. forces of 6 N southeast and 8 N northwest
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Part B
9N
4. Find the magnitude of the resultant of the four forces
shown in the given diagram.
7N
10 N
5N
៮៬ act at right angles to each other. Express the magnitude
5. Two forces ៮៬
F1 and F
2
៮៬
៮៬  and F
៮៬ .
F2 in terms of F
and direction of F1 ៮៬
1
2
6. Find the magnitude and the direction (to the nearest degree) of the resultant
of each of the following systems of forces.
a. forces of 3 N and 8 N acting at an angle of 60º to each other
b. forces of 15 N and 8 N acting at an angle of 130º to each other
7. Find the magnitude and direction of the equilibrant of each of the following
systems of forces.
a. forces of 32 N and 48 N acting at an angle of 90º to each other
b. forces of 16 N and 10 N acting at an angle of 10º to each other
Communication
Knowledge/
Understanding
8. Is it easier to pull yourself up doing chin-ups when your hands are 60 cm
apart or 120 cm apart? Explain your answer.
9. A mass of 10 kg is suspended from a ceiling by two cords that make angles
of 30º and 45º with the ceiling. Find the tension in each of the cords.
10. Two forces of equal magnitude act at 60º to each other. If their resultant has
a magnitude of 30 N, find the magnitude of the equal forces.
11. Which of the following sets of forces acting on an object could produce
equilibrium?
a. 5 N, 2 N, 13 N
c. 13 N, 27 N, 14 N
b. 7 N, 5 N, 5 N
d. 12 N, 26 N, 13 N
12. Three forces of 5 N, 7 N, and 8 N are applied to an object. If the object is in
a state of equilibrium
a. show how the forces must be arranged
b. calculate the angle between the lines of action of the 5 N and 7 N forces
13. A man weighing 70 kg lies in a hammock whose ropes make angles of
20º and 25º with the horizontal. What is the tension in each rope?
14. A steel wire 40 m long is suspended between two fixed points 20 m apart.
A force of 375 N pulls the wire down at a point 15 m from one end of the
wire. State the tension in each part of the wire.
142 C H A P T E R 4
15. An advertising sign is supported by a horizontal steel brace
extending at right angles from the side of a building, and by
a wire attached to the building above the brace at an angle
of 25º. If the force of gravity on the sign is 850 N, find the
tension in the wire and the compression in the steel brace.
25°
16. Find the x-and y-components of each of the vectors u៮៬, v៮៬,
៮៬.
and w
y
u =5
x
40°
65°
w = 12
v =9
17. A tractor is towing a log using a cable inclined at an angle of 15º to the
horizontal. If the tension in the cable is 1470 N, what is the horizontal force
moving the log?
18. A piece of luggage is on a conveyer belt that is inclined at an angle of 28º.
If the luggage has a mass of 20 kg
a. determine the components of the force of gravity parallel to and
perpendicular to the conveyer belt
b. explain the physical effect of each of these components
19. A child with a mass of 35 kg is sitting on a swing attached to a tree branch by
a rope 5 m in length. The child is pulled back 1.5 m measured horizontally.
a. What horizontal force will hold the child in this position?
b. What is the tension in the rope?
Application
20. The main rotor of a helicopter produces a force of 55 kN. If the helicopter
flies with the rotor revolving about an axis tilted at an angle of 8º to the
vertical
a. find the components of the rotor force parallel to and perpendicular
to the ground
b. explain the physical effect on the helicopter of each component of the
rotor force
21. In order to keep a 250-kg crate from sliding down a ramp inclined at 25º, the
force of friction that acts parallel to and up the ramp must have a magnitude
of at least how many newtons?
22. A lawn roller with a mass of 50 kg is being pulled with a force of 320 N.
If the handle of the roller makes an angle of 42º with the ground, what
horizontal component of the force is causing the roller to move?
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Part C
23. Three forces, each of which is perpendicular to the
other two, act on an object. If the magnitudes of
these forces are 6 N, 15 N, and 10 N, respectively,
find the magnitude and direction of the resultant.
(State the angles that the resultant makes with the
two larger forces.)
10 N
15 N
6N
24. Two tugs are towing a ship. The smaller tug is 10º off the port bow and the
larger tug is 20º off the starboard bow. The larger tug pulls twice as hard as
the smaller tug. In what direction will the ship move?
25. Braided cotton string will break when the tension exceeds 300 N. Suppose
that a weight of 400 N is suspended from a 200-cm length of string, the upper
ends of which are tied to a horizontal rod at points 120 cm apart.
a. Show that the string will support the
weight, when the weight is hung at the
centre of the string.
120 cm
100 cm
100 cm
400 N
b. Will the string break if the weight is 80 cm from one end of the string?
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