# Force and Mass by David Halliday and Robert Resnick

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/s2. Next we push on body X with the same force and find that it accelerates at 0.25 m/s2. Let's make the (correct) assumption that with the same force,

$\frac{{m}_{x}}{{m}_{0}}\frac{{a}_{0}}{{a}_{x}}$

and thus

${m}_{x}={m}_{0}\frac{{a}_{0}}{{a}_{x}}=\left(1.0\mathrm{kg}\right)\frac{1.0{\mathrm{m/s}}^{2}}{0.25{\mathrm{m/s}}^{2}}=4.0\mathrm{kg}$

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/s2) and then to body X (getting an acceleration of 2.0 m/s2). We would then calculate the mass of X as

${m}_{x}={m}_{0}\frac{{a}_{0}}{{a}_{x}}=\left(1.0\mathrm{kg}\right)\frac{8.0{\mathrm{m/s}}^{2}}{2.0{\mathrm{m/s}}^{2}}=4.0\mathrm{kg}$

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,

${\stackrel{\to }{F}}_{\mathrm{net}}=m\stackrel{\to }{a}$ (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 ${\stackrel{\to }{F}}_{\mathrm{net}}$ 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:

${\stackrel{\to }{F}}_{\mathrm{net,x}}={\mathrm{ma}}_{\mathrm{x,}}\phantom{\rule{1.5ex}{0ex}}{\stackrel{\to }{F}}_{\mathrm{net,y}}={\mathrm{ma}}_{\mathrm{y,}}\phantom{\rule{1.5ex}{0ex}}{\stackrel{\to }{F}}_{\mathrm{net,z}}={\mathrm{ma}}_{\mathrm{z,}}$

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 ax of the body's acceleration, but causes no acceleration in the y and z directions. Turned around, the acceleration component ax 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 $\stackrel{\to }{a}=0$. 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/s2) = 1 kg ˙ m/s2. (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/s2 CGS dyne gram(g) cm/ss2 British Pound(lb) slug ft/s2

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 ${\stackrel{\to }{F}}_{\mathrm{net}}$ 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, ${\stackrel{\to }{F}}_{\mathrm{net}}=m\stackrel{\to }{a}$, where m is the total mass of the system.