The Cause of Motion

Motion doesn’t happen without a cause. Newton’s second law builds on the first law by explaining how force causes motion to change. This relationship is not just conceptual—it’s quantitative.

In biology and health sciences, forces are everywhere. Muscle contractions apply forces to bones and tissues. Circulatory flow responds to pressure gradients (forces). Respiratory airflows are governed by pressure differences, and even cellular transport can involve physical forces.

Understanding Force and Systems

To apply Newton’s second law, we must first clarify some important terms:

• System: The object or group of objects we are analyzing.

• External force: A force acting on the system from outside its boundaries.

• Net force: The vector sum of all external forces acting on a system.

Example: Pushing a Wheelchair

If a nurse pushes a patient in a wheelchair, the system includes the patient and the chair. The force from the nurse’s hands is external. The force between the patient and the chair’s seat is internal and doesn’t affect the motion of the whole system.

For example, in Figure 25.1(a) the system of interest is the wagon plus the child in it. The two forces exerted by the other children are external forces. An internal force acts between elements of the system. Again looking at Figure 25.1(a), the force the child in the wagon exerts to hang onto the wagon is an internal force between elements of the system of interest. Only external forces affect the motion of a system, according to Newton’s first law. (The internal forces actually cancel, as we shall see in the next section.) You must define the boundaries of the system before you can determine which forces are external. Sometimes the system is obvious, whereas other times identifying the boundaries of a system is more subtle. The concept of a system is fundamental to many areas of physics, as is the correct application of Newton’s laws. This concept will be revisited many times on our journey through physics.

Now, it seems reasonable that acceleration should be directly proportional to and in the same direction as the net (total) external force acting on a system. This assumption has been verified experimentally and is illustrated in Figure 25.1. In part (a), a smaller force causes a smaller acceleration than the larger force illustrated in part (c). For completeness, the vertical forces are also shown; they are assumed to cancel since there is no acceleration in the vertical direction. The vertical forces are the weight [latex]\mathbf{\text{w}}[/latex] and the support of the ground [latex]\mathbf{\text{N}}[/latex], and the horizontal force [latex]\mathbf{\text{f}}[/latex] represents the force of friction. These will be discussed in more detail in later sections. For now, we will define friction as a force that opposes the motion past each other of objects that are touching. Figure 25.1(b) shows how vectors representing the external forces add together to produce a net force, [latex]{\mathbf{\text{F}}}_{\text{net}}[/latex].

To obtain an equation for Newton’s second law, we first write the relationship of acceleration and net external force as the proportionality

where the symbol [latex]\propto[/latex] means “proportional to,” and [latex]{\mathbf{\text{F}}}_{\text{net}}[/latex] is the net external force. (The net external force is the vector sum of all external forces and can be determined graphically, using the head-to-tail method, or analytically, using components. The techniques are the same as for the addition of other vectors, and are covered in Two-Dimensional Kinematics.) This proportionality states what we have said in words—acceleration is directly proportional to the net external force. Once the system of interest is chosen, it is important to identify the external forces and ignore the internal ones. It is a tremendous simplification not to have to consider the numerous internal forces acting between objects within the system, such as muscular forces within the child’s body, let alone the myriad of forces between atoms in the objects, but by doing so, we can easily solve some very complex problems with only minimal error due to our simplification

Now, it also seems reasonable that acceleration should be inversely proportional to the mass of the system. In other words, the larger the mass (the inertia), the smaller the acceleration produced by a given force. And indeed, as illustrated in Figure 25.2, the same net external force applied to a car produces a much smaller acceleration than when applied to a basketball. The proportionality is written as

where [latex]m[/latex] is the mass of the system. Experiments have shown that acceleration is exactly inversely proportional to mass, just as it is exactly linearly proportional to the net external force.

It has been found that the acceleration of an object depends only on the net external force and the mass of the object. Combining the two proportionalities just given yields Newton’s second law of motion.

Application: Weight as a Force

Weight is the force of gravity acting on an object. It is calculated using Newton’s second law, where the acceleration is the gravitational acceleration [latex]g = 9.80\ \text{m/s}^2[/latex]:

[latex]\mathbf{w} = m \mathbf{g}[/latex]

This means a 1.0-kg mass weighs:

[latex]w = (1.0\ \text{kg})(9.80\ \text{m/s}^2) = 9.8\ \text{N}[/latex]

Because [latex]g[/latex] can vary based on location (e.g., on the Moon or Mars), weight is not an intrinsic property of an object. Mass is, however—it remains the same anywhere in the universe.

Common Misconceptions: Mass vs. Weight

Mass and weight are often used interchangeably in everyday language. However, in science, these terms are distinctly different from one another. Mass is a measure of how much matter is in an object. The typical measure of mass is the kilogram (or the “slug” in English units). Weight, on the other hand, is a measure of the force of gravity acting on an object. Weight is equal to the mass of an object ([latex]m[/latex]) multiplied by the acceleration due to gravity ([latex]g[/latex]). Like any other force, weight is measured in terms of newtons (or pounds in English units).

Assuming the mass of an object is kept intact, it will remain the same, regardless of its location. However, because weight depends on the acceleration due to gravity, the weight of an object can change when the object enters into a region with stronger or weaker gravity. For example, the acceleration due to gravity on the Moon is [latex]1.67\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}[/latex] (which is much less than the acceleration due to gravity on Earth, [latex]9.80\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}[/latex]). If you measured your weight on Earth and then measured your weight on the Moon, you would find that you “weigh” much less, even though you do not look any skinnier. This is because the force of gravity is weaker on the Moon. In fact, when people say that they are “losing weight,” they really mean that they are losing “mass” (which in turn causes them to weigh less).

Take-Home Experiment: Mass and Weight

What do bathroom scales measure? When you stand on a bathroom scale, what happens to the scale? It depresses slightly. The scale contains springs that compress in proportion to your weight—similar to rubber bands expanding when pulled. The springs provide a measure of your weight (for an object which is not accelerating). This is a force in newtons (or pounds). In most countries, the measurement is divided by 9.80 to give a reading in mass units of kilograms. The scale measures weight but is calibrated to provide information about mass. While standing on a bathroom scale, push down on a table next to you. What happens to the reading? Why? Would your scale measure the same “mass” on Earth as on the Moon?

Example 25.1: What Acceleration Can a Person Produce when Pushing a Lawn Mower?

Suppose that the net external force (push minus friction) exerted on a lawn mower is 51 N (about 11 lb) parallel to the ground. The mass of the mower is 24 kg. What is its acceleration?

Strategy

Since [latex]{\mathbf{\text{F}}}_{\text{net}}[/latex] and [latex]m[/latex] are given, the acceleration can be calculated directly from Newton’s second law as stated in [latex]{\mathbf{\text{F}}}_{\text{net}}=m\mathbf{\text{a}}[/latex].

Solution

The magnitude of the acceleration [latex]a[/latex] is [latex]a=\frac{{F}_{\text{net}}}{m}[/latex]. Entering known values gives

[latex]a=\frac{\text{51 N}}{\text{24 kg}}[/latex]

Substituting the units [latex]\text{kg}\cdot {\text{m/s}}^{2}[/latex] for N yields

[latex]a=\frac{\text{51 kg}\cdot {\text{m/s}}^{2}}{\text{24 kg}}=\text{2.1 m}{\text{/s}}^{2}.[/latex]

Discussion

The direction of the acceleration is the same direction as that of the net force, which is parallel to the ground. There is no information given in this example about the individual external forces acting on the system, but we can say something about their relative magnitudes. For example, the force exerted by the person pushing the mower must be greater than the friction opposing the motion (since we know the mower moves forward), and the vertical forces must cancel if there is to be no acceleration in the vertical direction (the mower is moving only horizontally). The acceleration found is small enough to be reasonable for a person pushing a mower. Such an effort would not last too long because the person’s top speed would soon be reached.

Example 25.2 What Rocket Thrust Accelerates This Sled?

Prior to manned space flights, rocket sleds were used to test aircraft, missile equipment, and physiological effects on human subjects at high speeds. They consisted of a platform that was mounted on one or two rails and propelled by several rockets. Calculate the magnitude of force exerted by each rocket, called its thrust [latex]\mathbf{\text{T}}[/latex], for the four-rocket propulsion system shown in Figure 25.4. The sled’s initial acceleration is [latex]\text{49}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2},[/latex] the mass of the system is 2100 kg, and the force of friction opposing the motion is known to be 650 N.

Strategy

Although there are forces acting vertically and horizontally, we assume the vertical forces cancel since there is no vertical acceleration. This leaves us with only horizontal forces and a simpler one-dimensional problem. Directions are indicated with plus or minus signs, with right taken as the positive direction. See the free-body diagram in the figure.

Solution

Since acceleration, mass, and the force of friction are given, we start with Newton’s second law and look for ways to find the thrust of the engines. Since we have defined the direction of the force and acceleration as acting “to the right,” we need to consider only the magnitudes of these quantities in the calculations. Hence we begin with

[latex]{F}_{\text{net}}=\text{ma},[/latex]

where [latex]{F}_{\text{net}}[/latex] is the net force along the horizontal direction. We can see from Figure 25.4 that the engine thrusts add, while friction opposes the thrust. In equation form, the net external force is

[latex]{F}_{\text{net}}=4T-f.[/latex]

Substituting this into Newton’s second law gives

[latex]{F}_{\text{net}}=\text{ma}=4T-f.[/latex]

Using a little algebra, we solve for the total thrust 4T:

[latex]4T=\text{ma}+f.[/latex]

Substituting known values yields

[latex]4T=\text{ma}+f=\left(\text{2100 kg}\right)\left({\text{49 m/s}}^{2}\right)+\text{650 N}.[/latex]

So the total thrust is

[latex]4T=1.0×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N},[/latex]

and the individual thrusts are

[latex]T=\frac{1.0×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N}}{4}=2.6×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{N}.[/latex]

Discussion

The numbers are quite large, so the result might surprise you. Experiments such as this were performed in the early 1960s to test the limits of human endurance and the setup designed to protect human subjects in jet fighter emergency ejections. Speeds of 1000 km/h were obtained, with accelerations of 45 [latex]g[/latex]‘s. (Recall that [latex]g[/latex], the acceleration due to gravity, is [latex]9\text{.}{\text{80 m/s}}^{2}[/latex]. When we say that an acceleration is 45 [latex]g[/latex]‘s, it is [latex]\text{45}×9\text{.}{\text{80 m/s}}^{2}[/latex], which is approximately [latex]{\text{440 m/s}}^{2}[/latex].) While living subjects are not used any more, land speeds of 10,000 km/h have been obtained with rocket sleds. In this example, as in the preceding one, the system of interest is obvious. We will see in later examples that choosing the system of interest is crucial—and the choice is not always obvious.

Newton’s second law of motion is more than a definition; it is a relationship among acceleration, force, and mass. It can help us make predictions. Each of those physical quantities can be defined independently, so the second law tells us something basic and universal about nature. The next section introduces the third and final law of motion.