From everyday experience you already know that applying a given force to bodies (say, a baseball and a bowling ball) results in different accelerations. The common explanation is correct: The object with the larger mass is accelerated less. But we can be more precise. The acceleration is actually inversely related to the mass (rather than, say, the square of the mass).

Let's justify that inverse relationship. Suppose, as previously, we push on the standard body (defined
to have a mass of exactly 1 kg) with a force of magnitude 1 N. The body accelerates with a magnitude of
1 m/s^{2}. Next we push on body X with the same force and find that it accelerates at 0.25 m/s^{2}.
Let's make the (correct) assumption that with the same force,

and thus

${m}_{x}={m}_{0}\frac{{a}_{0}}{{a}_{\mathrm{x}}}=\left(1.0\mathrm{kg}\right)\frac{1.0{\mathrm{m/s}}^{2}}{0.25{\mathrm{m/s}}^{2}}=4.0\mathrm{kg}$Defining the mass of X in this way is useful only if the procedure is consistent. Suppose we apply an
8.0 N force first to the standard body (getting an acceleration of 8.0 m/s^{2}) and then to body
X (getting an acceleration of 2.0 m/s^{2}). We would then calculate the mass of X as

which means that our procedure is consistent and thus usable.

The results also suggest that mass is an intrinsic characteristic of a body - it automatically comes with the existence of the body. Also, it is a scalar quantity. However, the nagging question remains:What, exactly, is mass?

Since the word mass is used in everyday English, we should have some intuitive understanding
of it, maybe something that we can physically sense. Is it a body's size, weight, or density?
The answer is no, although those characteristics are sometimes confused with mass. We can
say only that *the mass of a body is the characteristic that relates a force on the body
to the resulting acceleration*. Mass has no more familiar definition; you can have a
physical sensation of mass only when you try to accelerate a body, as in the kicking of a
baseball or a bowling ball.

**Newton's Second Law**

All the definitions, experiments, and observations we have discussed so far can be summarized in one neat statement:

**• Newton's Second Law:** The net force on a body is equal to the product of the
body's mass and its acceleration.

In equation form,

${\overrightarrow{F}}_{\mathrm{net}}=m\overrightarrow{a}$ (Newton's second law). (5-1)

**Identify the Body.** This simple equation is the key idea for nearly all the
homework problems in this chapter, but we must use it cautiously. First, we must
be certain about which body we are applying it to. Then
${\overrightarrow{F}}_{\mathrm{net}}$
must be the vector sum of *all* the forces that act on *that* body. Only forces that act
on *that* body are to be included in the vector sum, not forces acting on other bodies that might be
involved in the given situation. For example, if you are in a rugby scrum, the net force on
*you* is the vector sum of all the pushes and pulls on *your* body. It does not include
any push or pull on another player from you or from anyone else.
Every time you work a force problem, your first step is to clearly state the body
to which you are applying Newton's law.

**Separate Axes.** Like other vector equations, Eq. 5-1 is equivalent to three
component equations, one for each axis of an xyz coordinate system:

Each of these equations relates the net force component along an axis to the
acceleration along that same axis. For example, the first equation tells us that
the sum of all the force components along the *x* axis causes the *x* component a_{x}
of the body's acceleration, but causes no acceleration in the *y* and *z* directions.
Turned around, the acceleration component a_{x} is caused only by the sum of the
force components along the *x* axis and is *completely* unrelated to force components
along another axis. In general,

**•** The acceleration component along a given axis is caused *only by* the sum of
the force components along that *same* axis, and not by force components along any other axis.

**Forces in Equilibrium.** Equation 5-1 tells us that if the net force on a body is
zero, the body's acceleration
$\overrightarrow{a}=0$. If the
body is at rest, it stays at rest; if it is moving, it continues to move at constant velocity.
In such cases, any forces on the body *balance* one another, and both the forces and the
body are said to be in *equilibrium*. Commonly, the forces are also said to *cancel*
one another, but the term "cancel" is tricky. It does *not* mean that the forces cease to
exist (canceling forces is not like canceling dinner reservations). The forces still act on the
body but cannot change the velocity.

**Units**. For SI units, Eq. 5-1 tells us that

1 N = (1 kg)(1 m/s^{2}) = 1 kg ˙ m/s^{2}. (5-3)

Some force units in other systems of units are given in Table 5-1.

**Diagrams**. To solve problems with Newton's second law, we often draw a *free-body
diagram* in which the only body shown is the one for which we are summing forces. A sketch
of the body itself is preferred by some teachers but, to save space, we shall usually represent
the body with a dot. Also, each force on the body is drawn as a vector arrow with its tail
anchored on the body.A coordinate system is usually included, and the acceleration of the body
is sometimes shown with a vector arrow (labeled as an acceleration). This whole procedure is
designed to focus our attention on the body of interest.

System | Force | Mass | Acceleration |

SI | newton(N) | Kilogram(kg) | m/s^{2} |

CGS | dyne | gram(g) | cm/ss^{2} |

British | Pound(lb) | slug | ft/s^{2} |

**External Forces Only.** A *system* consists of one or more bodies, and any
force on the bodies inside the system from bodies outside the system is called an
*external force*. If the bodies making up a system are rigidly connected to one another,
we can treat the system as one composite body, and the net force
${\overrightarrow{F}}_{\mathrm{net}}$
on it is the vector sum of all external forces. (We do not include internal forces-that is,
forces between two bodies inside the system. Internal forces cannot accelerate the system.)
For example, a connected railroad engine and car form a system. If, say, a tow line pulls on
the front of the engine, the force due to the tow line acts on the whole engine-car system.
Just as for a single body, we can relate the net external force on a system to its
acceleration with Newton's second law,
${\overrightarrow{F}}_{\mathrm{net}}=m\overrightarrow{a}$,
where m is the total mass of the system.

**About the Authors**

David Halliday was an American physicist known for his physics textbooks, *Physics and
Fundamentals of Physics*, which he wrote with Robert Resnick. Both textbooks have
been in continuous use since 1960 and are available in more than 47 languages.

Robert Resnick was a physics educator and author of physics textbooks. He was born in Baltimore, Maryland on January 11, 1923 and graduated from the Baltimore City College high school in 1939. He received his B.A. in 1943 and his Ph.D. in 1949, both in physics from Johns Hopkins University.

The 10th edition of Halliday's Fundamentals of Physics, Extended building upon previous issues by offering several new features and additions. The new edition offers most accurate, extensive and varied set of assessment questions of any course management program in addition to all questions including some form of question assistance including answer specific feedback to facilitate success. The text also offers multimedia presentations (videos and animations) of much of the material that provide an alternative pathway through the material for those who struggle with reading scientific exposition.

Furthermore, the book includes math review content in both a self-study module for more in-depth review and also in just-in-time math videos for a quick refresher on a specific topic. The Halliday content is widely accepted as clear, correct, and complete. The end-of-chapters problems are without peer. The new design, which was introduced in 9e continues with 10e, making this new edition of Halliday the most accessible and reader-friendly book on the market.

A Reader says,"As many reviewers have noted, this is a great physics book used widely in university technical programs as a first course in technical physics, with calculus. I find it is the one book I start with when trying to understand physical concepts at a useful but basic level. It has broad coverage and is well written . To go beyond this book requires specialized books on each topic of interest (electromagnetics, quantum mechanics, thermodynamics, etc.)."

Reader Frank says, "The treatment is sound, thorough, and clear. I've owned the early editions of Halliday and Resnick for years. I'm very happy that I updated my library with this 10th edition. The topics are covered in a very logical order. The study features and worked examples are outstanding. Don't hesitate to buy this book! Reading it is awesome on the Kindle app on the iPad."

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