Unit 6: Forces and Newton’s Laws of Motion
Lesson 6.3
Newton’s First Law of Motion
Contents
Introduction
1
Learning Objectives
2
Warm Up
2
Learn about It!
The First Law of Motion
Inertial Frames of Reference
Superposition of Forces
4
4
5
8
Key Points
16
Key Formulas
17
Check Your Understanding
18
Challenge Yourself
20
Bibliography
20
Key to Try It!
21
Unit 6: Forces and Newton’s Laws of Motion
Lesson 6.3
Newton’s First Law of Motion
Introduction
For many years, people held Aristotle’s position that the natural conclusion of any moving
object is to come to rest. While this might appeal suitably to the common sense, it took
thousands of years and the ingenuity of Galileo Galilei to disprove that such is not the case.
He conducted several experiments on balls and various surfaces that eventually led to the
conclusion that objects possess “inertia,” which is responsible for maintaining states of rest
or motion in things. In the lesson that follows, we will find out exactly how another great
mind, Sir Isaac Newton, has kept the ball rolling.
6.3. Newton’s First Law of Motion
1
Unit 6: Forces and Newton’s Laws of Motion
Learning Objectives
DepEd Competencies
In this lesson, you should be able to do the
following:
●
Explain
●
the
concept
of
inertial
●
Define inertial frames of reference
●
Apply Newton’s 1st Law to obtain
(STEM_GP12N-ld28).
qualitative and quantitative
reference frame.
conclusions about the contact and
Apply the law of inertia to describe the
in equilibrium
noncontact forces acting on a body
forces exerted on an object in
equilibrium.
●
Solve problems related to the law of
inertia.
(STEM_GP12N-le33).
●
Solve problems using Newton’s
Laws of Motion in contexts such
as, but not limited to, ropes and
pulleys, the design of mobile
sculptures, transport of loads on
conveyor belts, force needed to
move stalled vehicles, and
determination of safe driving
speeds on banked curved roads
(STEM_GP12N-le38).
Warm Up
Forces and Motion Basics
5 minutes
The following interactive will allow you to visualize the forces at work when doing an array of
actions. Here, we will be able to determine when forces are in equilibrium or not.
Material
●
laptop with internet connection
Procedure
1. Open the simulation. For 30 seconds, explore the controls and options available. A
screenshot of the simulator is presented in Fig. 6.3.1.
6.3. Newton’s First Law of Motion
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Unit 6: Forces and Newton’s Laws of Motion
Forces and Motion: Basics
University of Colorado Boulder, “Forces and Motion: Basics,” PhET
Interactive Simulations,
https://phet.colorado.edu/sims/html/forces-and-motion-basics/lates
t/forces-and-motion-basics_en.html, last accessed on March 28,
2020.
Fig. 6.3.1. Forces and Motion: Basics Simulator
2. Tick the options Force and Speed.
3. Apply a value of 100 N to the box. Refer to the speedometer and observe what
happens to the box’s speed.
4. Reset the simulation and tick the options Force and Speed again. Apply 100 N to the
box for ten seconds and slide the value of applied force back to zero. Observe what
happens.
Guide Questions
1. What happened to the speed of the box when you applied a force of 100 N to it?
2. What happened when you reduced the force to zero?
3. What does this conclude about the nature of motion?
6.3. Newton’s First Law of Motion
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Unit 6: Forces and Newton’s Laws of Motion
Learn about It!
It appears almost self-evident that when an object is motionless, no net force is acting upon
it. It truly is the case, but not always. Zero net force may actually act on an object even
when it is moving, as shown by the simulation that you have just interacted with. Let us try
to prove how this is possible.
How do forces acting on an object in equilibrium
behave?
The First Law of Motion
Consider a book being slid by a horizontal force across a horizontal, rough tabletop. After it
has been released, the book will slow down and eventually stop. It is then tempting to
conclude that when an object is in motion, it is natural for it to cease moving at some point
unless a force will continue to act upon it to keep it moving.
This, however, is one of the most common misconceptions related to force and motion.
Imagine, for instance, that the book is sliding—this time, along a polished, wet surface like a
slab of ice. After the book has been released, it will slide at a much farther distance before
going to a halt. What differs between the first situation and the second is the friction force
exerted by the two surfaces: the polished surface exerted less friction force than the rough
tabletop. Without friction, the book in both situations will simply keep on moving
indefinitely. Hence, to say that a force is needed to keep an object in motion is misleading
and untrue.
This proves that when no net force is acting upon an object, it will remain at rest, or it will
move forward with a constant velocity, such that no net force is required to sustain its
motion. This is known as the first law of motion.
6.3. Newton’s First Law of Motion
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Unit 6: Forces and Newton’s Laws of Motion
Remember
The first law of motion states that “every object continues in its
state of rest, or of uniform velocity in a straight line, as long as no
net force acts on it.”
This tendency of objects to remain in motion is referred to as inertia, which is also the
reason why Newton’s first law is also called the law of inertia. An inertial frame of
reference is a perspective considered to identify whether an object is at rest, or is moving
with a constant velocity.
Inertial Frames of Reference
The law of inertia cannot be applied consistently for every frame of reference. Recall from
your lesson on relative velocity that motion may be characterized differently according to
the location or perspective we are observing it from. It may initially be a perplexing idea to
wrap our heads around, so, let us simplify what an inertial frame of reference is, through
the following salient points:
1. It is a reference frame with constant velocity, i.e., it is at rest, or it is moving with a
constant speed in a linear direction.
2. It is not accelerating.
3. It is a reference frame in which the first law is applicable.
4. No fictitious forces arise from it.
Did You Know?
A fictitious force, also called d’Alembert force (named after the
French
mathematician
and
physicist
Jean-Baptiste
le
Rond
d’Alembert), is a force that seems to be acting upon an object
whose motion is characterized using a non-inertial reference frame,
i.e., accelerating or rotating.
6.3. Newton’s First Law of Motion
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Unit 6: Forces and Newton’s Laws of Motion
Remember
Any object viewed from an inertial frame of reference will obey
the first law and continue to have a constant velocity, unless acted
upon by a nonzero net force.
The scenario that follows will allow us to visualize the implications of inertial (and
non-inertial) frames of reference. Consider a passenger (1) wearing roller skates (2) in the
aisle of the bus that is (3) speeding up on a straight road, as shown in Fig. 6.3.2. (The
numbers indicate that the following conditions are essential in understanding the concept
of the frame of reference.) As the bus speeds up, the passenger will move backward with
respect to the bus (Fig. 6.3.2a). As it slows down to a halt, the passenger will move forward
relative to the vehicle (Fig. 6.3.2b). Here arises the second most common misconception
about motion: in both cases, it might appear as though no net force is acting upon the
passenger, but his velocity has changed, and this analysis is flawed.
Fig. 6.3.2. A passenger inside an accelerating vehicle
6.3. Newton’s First Law of Motion
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Unit 6: Forces and Newton’s Laws of Motion
Here, the bus’s acceleration is relative to Earth and cannot be applied as an inertial frame of
reference. An observer inside the bus’s reference frame might perceive as though a net
force has been exerted on the passenger since his velocity with respect to the bus has
continued to change. In point of fact, this net force is zero. The observer cannot apply the
law of inertia in the bus’s reference frame since it is not an inertial reference frame to begin
with. This can be reviewed from the four qualifications of inertial reference frame listed
above.
Did You Know?
The Earth is considered to be an approximate inertial reference
frame. Some extent of acceleration is observable because of the
planet’s rotation and revolution, but the effects are trivial and may
sometimes be ignored.
Under what conditions is an inertial frame of
reference applicable?
Other than Earth, frames of reference may be determined according to the relative
positions of objects with each other. The notion of relative velocity will tell us that if we have
one inertial frame A that obeys the first law, another frame B will also be an inertial
reference frame if it moves with respect to the motion of the first frame with a constant
velocity. This relation is mathematically expressed as follows:
Equation 6.3.1
where P is the object under consideration, A is the first inertial frame, B is the second inertial
frame,
frame B, and
is the constant velocity of the object P,
is the velocity of P relative to
is the velocity of frame B relative to A.
6.3. Newton’s First Law of Motion
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Unit 6: Forces and Newton’s Laws of Motion
Observers located separately in reference frames A and B will perceive a different velocity
for the object P. Equation 6.3.1, however, establishes mathematical proof that these two
observers will both perceive P to have a constant velocity or zero acceleration, and that no
net force has been exerted on P.
Remember
Reference frames are relative. Remember that if one frame is
considered inertial, another frame with respect to it with constant
velocity is also inertial.
Superposition of Forces
You have learned that motion and force are interrelated. When a force is exerted upon an
object, it is accompanied by a given result in motion.
Fig. 6.3.3. Superposition of forces
In the free-body diagram in Fig. 6.3.3 , two forces,
an object O. The diagram will tell us that
the forces exerted by
and
and
are acting simultaneously on
exerts a single force that is equal in amount to
. This is known as the superposition principle of forces
and establishes that the individual forces exerted at a point upon an object will produce a
similar effect as a single force equivalent to the resultant, or vector sum of the individual
forces. It is given by:
Equation 6.3.2
6.3. Newton’s First Law of Motion
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Unit 6: Forces and Newton’s Laws of Motion
where
is the net force exerted upon an object,
is the vector sum/resultant of the
individual forces, and F is the individual force acting on an object.
When forces are acting on both the x- and y-components, the x-component of the resultant
is the sum of all the forces along the x. The same is true for the y-component, thus:
and
The Pythagorean theorem may then be used to determine the magnitude of the net force,
as follows:
Equation 6.3.3
where
is the net force exerted upon an object,
x-axis, and
The angle
is the summation of forces along the
is the summation of forces along the y-axis.
is determined by the tangent function:
.
Remember
The angle
and
may be found in any of the four quadrants and
may be positive, negative, or zero. Be mindful of the signs
when dealing with each component.
6.3. Newton’s First Law of Motion
9
Unit 6: Forces and Newton’s Laws of Motion
Now, let us try to prove the first law of motion using the superposition principle. Consider
the book that we described earlier sliding on a polished wet surface. If the book is initially at
rest and a single horizontal force
is exerted upon it, the book will be set in motion. Here,
the net force is nonzero, and is equal to
, since the other two forces, weight and normal
force are equal and will cancel each other. If a second force
which is equal in magnitude but opposite in direction with
will be exerted on the book,
, their resultant will be zero.
This shows that zero net force implies that there is no force at all, that is, the object is either
at rest or moving in a straight line with constant speed. Objects like these are considered to
be in equilibrium. Newton’s first law of motion is thus mathematically expressed as follows:
Equation 6.3.4
where
is the net force exerted upon an object.
Remember
For an object to be in equilibrium, either no force must act upon it,
or the resultant of the individual forces exerted upon it must have a
net force of zero.
What does the first law of motion state?
Let’s Practice!
6.3. Newton’s First Law of Motion
10
Unit 6: Forces and Newton’s Laws of Motion
Example 1
A 0.020-kilogram block of wood with a constant velocity slides on a rough horizontal
tabletop. If there is an external force of 2.50 newtons exerted upon the block of wood,
calculate the magnitude of the friction force acting on it.
Solution
Step 1:
Identify what is required in the problem.
You are asked to solve the magnitude of the friction force exerted on the block of
wood.
Step 2:
Identify the given in the problem.
The mass (0.020 kg) and the force (2.50 N) are given.
Step 3:
Draw the free-body diagram of the situation.
The law of inertia would posit that since the block of wood is moving with
constant velocity, it has no acceleration if the friction force (f) has a similar
6.3. Newton’s First Law of Motion
11
Unit 6: Forces and Newton’s Laws of Motion
magnitude with the external force (F).
Observe from the free-body diagram that the external force (F) is in an
opposite direction from the friction force (f).
Step 4:
Write the working equation.
, thus:
Step 5:
Substitute the given values and find the answer.
Thus, the magnitude of the friction force acting on the block of wood is 2.50 N.
1 Try It!
A book that weighs .050 kg has been pushed at a constant velocity across the floor. If
there is an external force of 1.75 newtons exerted upon the book, calculate the
magnitude of the friction force acting on it.
Example 2
A 180-pound cargo is found inside a freight elevator that heads upward at a constant
velocity. The rope of the elevator snapped which caused it to fall. Determine the (a) weight
of the cargo in N, and (b) the normal force exerted by the elevator’s floor to the cargo before
and after the rope broke off.
Solution
Step 1:
Identify what is required in the problem.
You are asked to calculate the normal force exerted by the elevator floor to the
cargo before N1 and after N2 the rope broke.
Step 2:
Identify the given in the problem.
The mass (180 lbs) is given.
6.3. Newton’s First Law of Motion
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Unit 6: Forces and Newton’s Laws of Motion
Step 3:
Draw the free-body diagram for the situation.
Step 4:
Write the working equations.
First, convert 180 lbs to kg and solve for W.
Determine the value of N1.
, thus:
Solve N2.
After the rope snapped, both the cargo and the elevator were free-falling.
Hence, no normal force N2 was acting upon them.
Step 5:
Substitute the given value(s) and find the answers.
6.3. Newton’s First Law of Motion
13
Unit 6: Forces and Newton’s Laws of Motion
Thus, the weight (W) of the cargo in newtons is 800 N. The normal force acting on it, by
virtue of the law of inertia is also 800 N. After the elevator rope broke, the normal force
became 0 N.
2 Try It!
Calculate the (a) weight of the cargo in N, and (b) the normal force exerted by the
elevator’s floor to the cargo before and after the rope broke off if the mass of the
cargo in Example #2 is changed to 215 lbs.
Example 3
A 250 lb metal slab needed to be moved up inside a vehicle using a smooth inclined plane
with the following dimensions: l = 0.7 m, H = 0.3 m. What is the magnitude of the force
required to move the slab with a constant velocity? Ignore the effects of friction.
Solution
Step 1:
Identify what is required in the problem.
You are asked to calculate the magnitude of force required to move the slab up
the
incline.
Step 2:
Identify the given in the problem.
The weight of the metal slab (250 lbs) and the dimensions of the inclined plane
(l = 0.7 m and H = 0.3 m) are given.
6.3. Newton’s First Law of Motion
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Unit 6: Forces and Newton’s Laws of Motion
Step 3:
Draw the free-body diagram for the situation.
The law of inertia would posit that the metal slab will head upward, and the
force exerted upon it will be equal to the horizontal component of the weight
.
Step 4:
Write the working equations and substitute the given values.
First, convert 250 lbs to N.
Determine the angle of inclination.
Hypotenuse = 0.7 m
Opposite = 0.3 m
Solve Wx.
6.3. Newton’s First Law of Motion
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Unit 6: Forces and Newton’s Laws of Motion
Apply the law of inertia.
, thus:
Step 5:
Find the answer.
Thus, the magnitude of force required to move the slab up the inclined plane is 478.29 N.
3 Try It!
A 300 lb rock has to be moved up inside a vehicle using a smooth inclined plane with
the following dimensions: l = 0.8 m, H = 0.5 m. What is the magnitude of the force
required to move the slab?
Key Points
___________________________________________________________________________________________
●
The first law of motion states that “every object continues in its state of rest, or of
uniform velocity in a straight line, as long as no net force acts on it”.
●
The tendency of objects to remain in motion is referred to as inertia, which is also
the reason why Newton’s first law is also called the law of inertia.
●
An inertial frame of reference is a perspective considered to identify whether an
object is at rest, or is moving with a constant velocity.
●
Any object viewed from an inertial frame of reference will obey the first law and
continue to have a constant velocity unless acted upon by a nonzero net force.
●
For an object to be in equilibrium, either no force must act upon it, or the
resultant of the individual forces exerted upon it must have a net force of zero.
___________________________________________________________________________________________
6.3. Newton’s First Law of Motion
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Unit 6: Forces and Newton’s Laws of Motion
Key Formulas
___________________________________________________________________________________________
Concept
Formula
Description
Superposition of
Forces
Use this equation to
calculate the net force
where
is the net force exerted
●
upon an object;
exerted on an object when
all the forces are found in a
single dimension.
is the vector sum/
●
resultant of the individual
forces, and
●
F is the individual force
acting on an object.
Use the Pythagorean
theorem to solve the net
where
●
R is the net force exerted
upon an object;
●
Rx is the summation of
forces along the x-axis, and
●
force exerted on an object
when the individual forces
are located along the x- and
y-axes.
Ry is the summation of
forces along the y-axis.
The angle
function
6.3. Newton’s First Law of Motion
is determined by the
.
17
Unit 6: Forces and Newton’s Laws of Motion
Check Your Understanding
A.
Identify whether each statement is true or false.
______________
1. The first law of motion states that an object will continue to
be at rest or to move at a constant rate until a net force
acts on it.
______________
2. An inertial frame of reference may be accelerating.
______________
3. It is impossible for an object to have zero net force when it
is in motion.
______________
4. A relative frame of reference is a perspective considered
to identify whether an object is at rest, or is moving with
a constant velocity.
______________
5. Earth may be considered as an inertial reference frame.
______________
6. If one reference frame is considered inertial, another
frame with respect to it with constant velocity is also
inertial.
______________
7. The first law of motion is also called the law of momentum.
______________
8. The
superposition
principle
of
forces
is
given
by
.
______________
9. The tendency of objects to remain in motion is also called
inertia.
______________
6.3. Newton’s First Law of Motion
10. Objects that are in motion can never achieve equilibrium.
18
Unit 6: Forces and Newton’s Laws of Motion
B.
Answer the following questions.
A 50.0 kg athlete is suspended motionless from the end of a light rope attached to
the ceiling.
1. Draw a free-body diagram for the athlete.
2. Draw a free-body diagram for the rope.
3. Determine the athlete’s weight.
4. What are the magnitude and direction of the force that the rope has exerted
upon the athlete?
5. Calculate the tension force found on the upper end of the rope.
Two ropes of negligible mass hold a large steel ball of mass = 4.09 ✕ 106 g such that
it is motionless, as shown below.
1. Draw a free-body diagram for the ball.
2. Calculate the weight of the ball in N.
3. Identify the
along the vertical.
4. Determine the tension force TB in the rope with a 45-degree angle from the
vertical.
5. Determine the tension force TA in the horizontal rope.
6.3. Newton’s First Law of Motion
19
Unit 6: Forces and Newton’s Laws of Motion
Challenge Yourself
Answer the following questions. Provide mathematical proofs wherever necessary.
1. Why is the Earth considered only an approximate reference frame?
2. In a science fiction film, a spacecraft is travelling in outer space distance from
stars and planets when all of a sudden, its engine dies and the spacecraft slows
down and halts. Infer whether or not this agrees with the law of inertia.
3. You are riding a race car at a constant speed of 225 km/h when you pass by
another vehicle with a constant speed of 88 km/h. Which of the two cars has a
greater net force and why?
4. Is it possible for an object to achieve equilibrium when there is only one force
acting on it? Explain.
5. Consider a projectile thrown upward. Once it reaches its peak, it achieves a
velocity of zero. Does this imply that the projectile is in equilibrium at this point?
Bibliography
Faughn, Jerry S. and Raymond A. Serway. Serway’s College Physics (7th ed). Singapore:
Brooks/Cole, 2006.
Giancoli, Douglas C. Physics Principles with Applications (7th ed). USA: Pearson Education,
2014.
Macalalad, E. P. and Vergara, R. L. 2011. Exploring the Realms of Science: Physics. Valenzuela
City: JO-ES Publishing House, Inc.
Serway, Raymond A. and John W. Jewett, Jr. Physics for Scientists and Engineers with Modern
Physics (9th ed). USA: Brooks/Cole, 2014.
6.3. Newton’s First Law of Motion
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Unit 6: Forces and Newton’s Laws of Motion
Young, Hugh D., Roger A. Freedman, and A. Lewis Ford. Sears and Zemansky’s University
Physics with Modern Physics (13th ed). USA: Pearson Education, 2012.
Key to Try It!
1. 1.75 N
2. W = 956.58 N; N1 = 956.58 N; N2 = 0 N
3. 834.23 N
6.3. Newton’s First Law of Motion
21