Force Vectors by David Halliday and Robert Resnick

Newton's First Law

Before Newton formulated his mechanics, it was thought that some influence, a "force," was needed to keep a body moving at constant velocity. Similarly, a body was thought to be in its "natural state" when it was at rest. For a body to move with constant velocity, it seemingly had to be propelled in some way, by a push or a pull. Otherwise, it would "naturally" stop moving.

These ideas were reasonable. If you send a puck sliding across a wooden floor, it does indeed slow and then stop. If you want to make it move across the floor with constant velocity, you have to continuously pull or push it.

Send a puck sliding over the ice of a skating rink, however, and it goes a lot farther. You can imagine longer and more slippery surfaces, over which the puck would slide farther and farther. In the limit you can think of a long, extremely slippery surface (said to be a frictionless surface), over which the puck would hardly slow. (We can in fact come close to this situation by sending a puck sliding over a horizontal air table, across which it moves on a film of air.)

From these observations, we can conclude that a body will keep moving with constant velocity if no force acts on it. That leads us to the first of Newton's three laws of motion:

• Newton's First Law: If no net force acts on a body , the body's velocity cannot change; that is, the body cannot accelerate.

Before we begin working problems with forces, we need to discuss several features of forces, such as the force unit, the vector nature of forces, the combining of forces, and the circumstances in which we can measure forces (without being fooled by a fictitious force).

Figure 5-1 A force $\stackrel{\to }{F}$ on the standard kilogram gives that body an acceleration $\stackrel{\to }{a}$.

We can define the unit of force in terms of the acceleration a force would give to the standard kilogram, which has a mass defined to be exactly 1 kg. Suppose we put that body on a horizontal, frictionless surface and pull horizontally (Fig. 5-1) such that the body has an acceleration of 1 m/s2. Then we can define our applied force as having a magnitude of 1 newton (abbreviated N).

If we then pulled with a force magnitude of 2 N, we would find that the acceleration is 2 m/s2. Thus, the acceleration is proportional to the force. If the standard body of 1 kg has an acceleration of magnitude a (in meters per second per second), then the force (in newtons) producing the acceleration has a magnitude equal to a. We now have a workable definition of the force unit.

Vectors. Force is a vector quantity and thus has not only magnitude but also direction. So, if two or more forces act on a body, we find the net force (or resultant force) by adding them as vectors. A single force that has the same magnitude and direction as the calculated net force would then have the same effect as all the individual forces. This fact, called the principle of superposition for forces, makes everyday forces reasonable and predictable. The world would indeed be strange and unpredictable if, say, you and a friend each pulled on the standard body with a force of 1 N and somehow the net pull was 14 N and the resulting acceleration was 14 m/s2.

Forces are often represented with a vector symbol such as $\stackrel{\to }{F}$ and a net force is represented with the vector symbol ${\stackrel{\to }{F}}_{\mathrm{net}}$. As with other vectors, a force or a net force can have components along coordinate axes. When forces act only along a single axis, they are single-component forces. Then we can drop the overhead arrows on the force symbols and just use signs to indicate the directions of the forces along that axis.

The First Law. Instead of our previous wording, the more proper statement of Newton's First Law is in terms of a net force:

• Newton's First Law: If no net force acts on a body , the body's velocity cannot change; that is, the body cannot accelerate.

There may be multiple forces acting on a body, but if their net force is zero, the body cannot accelerate. So, if we happen to know that a body's velocity is constant, we can immediately say that the net force on it is zero.

Inertial Reference Frames

Newton's first law is not true in all reference frames, but we can always find reference frames in which it (as well as the rest of Newtonian mechanics) is true. Such special frames are referred to as inertial reference frames, or simply inertial frames.

An inertial reference frame is one in which Newton's laws hold.

For example, we can assume that the ground is an inertial frame provided we can neglect Earth's astronomical motions (such as its rotation).

Figure 5-2 (a) The path of a puck sliding from the north pole as seen from a stationary point in space. Earth rotates to the east. (b) The path of the puck as seen from the ground.

That assumption works well if, say, a puck is sent sliding along a short strip of frictionless ice - we would find that the puck's motion obeys Newton's laws. However, suppose the puck is sent sliding along a long ice strip extending from the north pole (Fig. 5-2a). If we view the puck from a stationary frame in space, the puck moves south along a simple straight line because Earth's rotation around the north pole merely slides the ice beneath the puck. However, if we view the puck from a point on the ground so that we rotate with Earth, the puck's path is not a simple straight line. Because the eastward speed of the ground beneath the puck is greater the farther south the puck slides, from our ground-based view the puck appears to be deflected westward (Fig. 5-2b).

However, this apparent deflection is caused not by a force as required by Newton's laws but by the fact that we see the puck from a rotating frame. In this situation, the ground is a noninertial frame, and trying to explain the deflection in terms of a force would lead us to a fictitious force. A more common example of inventing such a nonexistent force can occur in a car that is rapidly increasing in speed. You might claim that a force to the rear shoves you hard into the seat back.

We usually assume that the ground is an inertial frame and that measured forces and accelerations are from this frame. If measurements are made in, say, a vehicle that is accelerating relative to the ground, then the measurements are being made in a noninertial frame and the results can be surprising.

Which of the figure's six arrangements correctly show the vector addition of forces ${\stackrel{\to }{F}}_{1}$ and ${\stackrel{\to }{F}}_{2}$ to yield the third vector, which is meant to represent their net force ${\stackrel{\to }{F}}_{\mathrm{net}}$? c, d, and e (${\stackrel{\to }{F}}_{1}$ and ${\stackrel{\to }{F}}_{2}$ must be head to tail, ${\stackrel{\to }{F}}_{\mathrm{net}}$ must be from tail of one of them to head of the other)