LECTURE NOTES IN ENGINEERING MECHANICS

CHAPTER 5 – FORCE, MASS, AND ACCELERATION

5.1 Types of Forces

In general, force is considered as a “push” or “pull” exerted by one body on another.
Force can affect both the motion and the deformation of the body on which it acts.
Forces may arise from direct contact between bodies, or they may be applied at a
distance such as gravitational attraction. Contact forces are distributed over a
surface area of the body, sometimes the area over which a contact force is applied is
so small that it may be approximated by a point, in which case the force is said to be
concentrated force. The point of contact is also called the point of application of the
force.

5.2 Components of a force

Forces acting at some angle from the coordinate axes can be resolved into mutually
perpendicular forces called components.
y v
Fv
Fy F

θ β Fx
x

Fu u

Components of the force along the x-y axes

F x =Fcos ( θ ) ∧F y =Fsin ( θ )

Components of the force along the u-v axes

F u=Fcos ( β )∧F v =Fsin ( β )

5.3 Gravitational Force

Near to the surface of Earth a body of mass m experiences a force of gravity given by

W =mg

where g = 9.81 m/s2 and the force mg is directed towards the center of Earth.

5.4 Normal force

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Due to the gravitational force acting on a body its tendency is to accelerate towards
the center of Earth. This tendency is resisted when the body comes in contact with
the surface of another body. The component of the force normal (perpendicular) to
the plane of the surface is called the normal force, and is often represented by N.

Weight, W

Weight, W

⨂
⨂  

Normal force, N
Normal force, N

5.5 Force due to tension in strings

Ropes and strings exert forces due to tension in them. In most of discussions we will
assume the mass of the rope to be negligible in comparison to the masses of the
moving bodies. That is we pretend the strings to be of zero mass. Whenever a rope
is involved, the direction of the force due to the pull on the rope acts exactly in the
direction along the rope. The force with which we pull on the massless rope is
transmitted through the entire rope unchanged. The magnitude of this force is
referred to as the tension in the rope. Every rope can withstand only a certain
maximum force, but for now we will assume that all applied forces are below this
limit. Ropes cannot support a compression force.

Tension, T
                        
  
               

    ⨂

Weight, W

5.6 Force of friction

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While two solid surfaces are in contact, the force of friction is the force that resists
the tendency of the surfaces to move relative to each other in the lateral direction
(parallel to the surface). It acts in the direction opposite to the direction of tendency
of motion.

Weight, W

Weight, W

⨂
⨂  

Friction force, F
Friction force, F
Normal force, N
Normal force, N

For statics condition,

F f ≤( F max=μ s N )

For dynamic condition,

F f =μ k N

Where: μs = coefficient of statics friction

μk = coefficient of statics friction
N = Normal force

5.7 Laws of motion

Law of inertia: Newton’s first law of motion

The concept of inertia is the content of Newton’s first law of motion. It states that, a
body will maintain constant velocity, unless the net force on the body is non-zero. It
is also called the law of inertia. Velocity being a vector, constant here means
constant magnitude and constant direction. In other words, a body will move along
a straight line, unless acted upon by a force. An object at rest is said to be in static

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equilibrium. An object moving with constant velocity is said to be in dynamic

equilibrium.

Newton’s second law of motion

The first law of motion states that a force causes a change in velocity of the body. In
the second law the change in velocity is associated to the acceleration of the body.
Newton’s second law of motion states that for a fixed force the acceleration is
inversely proportional to the mass of the body. In this sense mass is often associated
to the notion of inertia, because mass resists change in velocity. Newton’s second
law of motion is expressed using the equation

∑ F=ma

where m is the mass of the body and a is the acceleration of the body.

Sign convention:
Positive force Negative force

F a
F a

force is toward the motion force is against the motion

Newton’s third law of motion

A force is exerted by one mass on another mass. Newton’s third law states that the
other mass exerts an equal and opposite reaction force on the first mass. Newton’s
Third Law is a consequence of the requirement that internal forces that is, forces
that act between different components of the same system must add to zero;
otherwise, their sum would contribute to a net external force and cause
acceleration, according to Newton’s Second Law.

SAMPLE PROBLEM 5.1

A mass m1 = 20.0 kg on a frictionless ramp is attached to a light string. The string
passes over a frictionless pulley and is attached to a hanging mass, m2. The ramp is
at an angle of θ = 30° above the horizontal. The mass m1 moves up the ramp
uniformly (at constant speed). Find the value of m2.

 
 
m1

m2

θ
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Solution:
The speed of the block is constant, thus, its acceleration is zero.
Considering the Free body diagram of block 1,
and applying the First Law of Motion

y
W1
T T
Considering the Free body diagram of block 2, θ
and applying the First Law of Motion
m1
x
m2
N1
θ
W2

SAMPLE PROBLEM 5.2

An elevator cabin has a mass of 360 kg, and the combined mass of the people inside
the cabin is 250 kg. The cabin is pulled upward by a cable, with a constant
acceleration of 4 m/s2. What is the tension in the cable?

Solution:
                          

Considering the Free body diagram shown, T

and applying the Second Law of Motion
av

   
            
                
       
W1
            

W2

SAMPLE PROBLEM 5.3

A block of mass m1 = 22 kg is at rest on a plane inclined at θ = 30.0° above the
horizontal. The block is connected via a rope and massless pulley system to another
block of mass m2 = 25 kg, as shown in the figure. The coefficients of kinetic friction

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between block 1 and the inclined plane is μk = 0.086. If the blocks are released from
rest, what is the displacement of block 2 in the vertical direction after 1.5 s?
 
 
m1

m2

Solution:
Considering the free body diagram for block 1, and
applying the Second Law of Motion along the y-axis

The acceleration along the y-axis is zero, thus,

y
W1 a
T T
θ

m1 F
x a
m2
N1
For dynamic condition, θ
W2

Applying the Second Law of Motion along the x-axis

Considering the Free body diagram of block 2,

and applying the second Law of Motion along vertical

Equate equation 1 and 2

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Displacement of block 2 in the vertical direction after 1.5 s

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PROBLEMS - SET 5
Instruction: Do as required. Present in writing the full details of your answer or
solutions. The grades you earn for each item will be based on correctness,
completeness and clarity of presentation.

PROBLEM 5.1
1. A 8 kg mass and a 20 kg mass are tied to a light string and hung over a
frictionless pulley. What is their acceleration?
a. 3.915 m/s2 c. 4.315 m/s2
b. 4.204 m/s 2
d. 3.395 m/s2

PROBLEM 5.2
2. An unknown mass and a 12 kg mass are tied to a light string and hung over a
frictionless pulley. If the tension in the string is 50 N, what is the unknown
mass?
a. 3.24 kg c. 5.24 kg
b. 4.24 kg d. 6.24 kg

PROBLEM 5.3
3. A lady pulls a cart with a force of 1800 N. Neglecting friction, if the cart changes
from resting to a speed of 1.25 m/s in a distance of 0.2 m, what is the total mass
of the cart?
a. 715 kg c. 480 kg
b. 460 kg d. 220 kg

PROBLEM 5.4
4. A man sees a 45 kg cart about to bump into a wall at 1.7 m/s. If the cart is 0.2 m
from the wall when he grabs it, how much force must he apply to stop it before
it hits?
a. 345.125 N c. 355.125 N
b. 335.125 N d. 325.125 N

PROBLEM 5.5
5. A man is pulling a 28 kg cart with a force of 100 N. Neglecting friction, how
much time does it take to get the cart from rest up to 2 m/s?
a. 0.66 s c. 0.76 s
b. 0.56 s d. 0.86 s

PROBLEM 5.6
6. A box is sliding down a ramp with an acceleration of 3 m/s 2. If the ramp is at an
angle of 25° relative to the ground, what is the coefficient of kinetic friction
between the box and the ramp?
a. 0.149 c. 0.139
b. 0.159 d. 0.129
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PROBLEM 5.7
7. A shipping crate that weighs 300 N is initially stationary on a loading dock. A
forklift arrives and lifts the crate with an upward force of 520 N, accelerating
the crate upward. What is the magnitude of the force due to gravity acting on
the shipping crate while it is accelerating upward?
a. 18 m/s2 c. 19 m/s2
b. 17 m/s2 d. 16 m/s2

PROBLEM 5.8
The gravitational acceleration on the Moon is a sixth of that on Earth. The weight of
an apple is 1.00 N on Earth.
8. What is the weight of the apple on the Moon?
a. 6 N c. 6 lb
b. 1/6 N d. 1/6 lb
9. What is the mass of the apple?
a. 6 kg c. 0.102 kg
b. 1/6 kg d. 0.202 kg

PROBLEM 5.9
10. A 430 N force accelerates a cart from 10 m/s to 18 m/s in 5 s. What is the mass
of the cart?
a. 278.75 kg c. 288.75 kg
b. 268.75 kg d. 298.75 kg

PROBLEM 5.10
11. An elevator cabin has a mass of 380 kg, and the combined mass of the people
inside the cabin is 180 kg. The cabin is pulled upward by a cable, in which there
is a tension force of 7600 N. What is the acceleration of the elevator?
a. 3.76 m/s2 c. 3.76 m/s2
b. 3.76 m/s2 d. 3.76 m/s2

PROBLEM 5.11
12. Four weights, of masses m1 = 6.50 kg, m2 = 3.80 kg, m3 = 10.50 kg, and m4 = 4.50
kg, are hanging from a ceiling as shown in the figure. They are connected with
ropes. What is the tension in the rope connecting masses m1 and m2?

m1
a. 184.43 N
b. 184.43 N
c. 184.43 N m2
d. 184.43 N
  m3

  m4
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PROBLEM 5.12
A hanging mass, M1 = 3.5 kg, is attached by a light string that runs over a frictionless
pulley to the front of a mass M2 = 1.50 kg that is initially at rest on the table. A third
mass M3 = 2.50 kg, which is also initially at rest on the same table, is attached to the
back of M2 by a light string. The coefficient of kinetic friction of all contact surfaces is
0.1.
13. Find the tension in the string connecting masses M 1 and M2.
a. 247.21 N c. 217.21 N
b. 237.21 N d. 227.21 N
14. Find the tension in the string connecting masses M2 and M3.
a. 50.82 N c. 40.82 N
b. 60.82 N d. 30.82 N

PROBLEM 5.13
15. A hanging mass, M1 = 0.5 kg, is attached by a light string that runs over a
frictionless pulley to a mass M2 = 1.5 kg that is initially at rest on a frictionless
ramp. The ramp is at an angle of θ = 30.0° above the horizontal and the pulley is
at the top of the ramp. Find the magnitude and direction of the acceleration of
M 1.
a. 4.905 m/s2, upward c. 4.905 m/s2, upward
b. 4.805 m/s2, downward d. 4.805 m/s2, downward

PROBLEM 5.14
Arriving on a newly discovered planet, the captain of a spaceship performed the
following experiment to calculate the gravitational acceleration for the planet: She
placed masses of 100 g and 200 g on an Atwood device made of massless string and
a frictionless pulley and measured that it took 1.5 seconds for each mass to travel 1
m from rest.
16. What is the gravitational acceleration for the planet?
a. 2.867 m/s2 c. 2.967 m/s2
2
b. 2.767 m/s d. 2.667 m/s2
17. What is the tension in the string?
a. 0.356 N c. 0.366 N
b. 0.346 N d. 0.376 N

PROBLEM 5.15
18. A crate slides down an inclined plane without friction. If it is released from rest
and reaches a speed of 6 m/s after sliding a distance of 2.5 m, what is the angle
of inclination of the plane with respect to the horizontal?
a. 43.22˚ c. 45.22˚
b. 41.22˚ d. 47.22˚

PROBLEM 5.16
19. Three objects with masses m1 = 36.5 kg, m2 = 19.2 kg, and m3 = 12.5 kg are
hanging from ropes that run over pulleys. What is the acceleration of m1?
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a. 0.69 m/s2 c. 0.79 m/s2

b. 0.89 m/s2     c. 0.59 m/s2
   
       

       
   
m2 m1 m3

PROBLEM 5.17
A bowling ball of mass M1 = 6 kg is initially at rest on the sloped side of a wedge of
mass M2 = 9 kg that is on a frictionless horizontal floor. The side of the wedge is
sloped at an angle of θ = 37° above the horizontal.

M1
P
M2
θ

20. What is the magnitude of the horizontal force P that should be exerted to keep
the ball at a constant height on the slope?
a. 135.89 N c. 155.89 N
b. 110.89 N d. 125.89 N
21. What is the magnitude of the acceleration of the wedge, if no external force is
applied?
a. 3.2433 m/s2 c. 3.1433 m/s2
2
b. 7.5346 m/s d. 7.8346 m/s2

PROBLEM 5.18
22. An engine block of mass M is on the flatbed of a pickup truck that is traveling in
a straight level road with an initial speed of 32 m/s. The coefficient of static
friction between the block and the bed is μs = 0.5. Find the minimum distance in
which the truck can come to a stop without the engine block sliding toward the
cab.
a. 114.38 m c. 124.38 m
b. 104.38 m d. 134.38 m

PROBLEM 5.19

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23. A skier starts with a speed of 2 m/s and skis straight down a slope with an
angle of 15˚ relative to the horizontal. The coefficient of kinetic friction between
her skis and the snow is 0.1. What is her speed after 10 s?
a. 16.91 m/s2 c. 18.91 m/s2
b. 17.91 m/s 2
d. 19.91 m/s2
PROBLEM 5.20
42. A wedge of mass m = 36 kg is located on a plane that is inclined by an angle θ =
22° with respect to the horizontal. A force F = 300 N in the horizontal direction
pushes on the wedge, as shown in the figure. The coefficient of kinetic friction
between the wedge and the plane is 0.16. What is the acceleration of the wedge
along the plane?

a. 1.596 m/s2
b. 2.596 m/s2 F m
c. 3.596 m/s2
d. 4.596 m/s2
θ

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