Dynamics

The study of the causes of motion is called "dynamics". We will introduce five new "dynamical" variables: momentum, force, potential energy, pressure and power. Each of these quantities will directly or indirectly involve the mass of the object. In fact, mass is defined dynamically in terms of force and acceleration, as we will see shortly.

First, let us define the momentum P:

P = m V.

Here, P is a vector, with separate x and y components. The above equation is really two equations, as we discussed earlier. Mass is a "scalar" quantity: it has only one component, and no direction. Its value is independent of the coordinate system we choose. By using the mass as a factor in the momentum, we give the momentum an additional aspect: objects may have higher momentum because they move faster, or because they are more massive. This agrees with our common sense notion of momentum as the quality of something which keeps it moving. The more momentum a thing possesses, the harder it is to halt its motion.

Pursuing our common sense a little further, we define the force as the time rate of change of momentum:

F = dP / dt.

That is, force is required to change momentum; alternatively, a change of momentum ("impulse") causes a force to be felt. Again, force is a vector quantity, and the above equation has an x and a y component. Since momentum has dimensions of mass times velocity, force has dimensions of mass times acceleration. In SI units, the unit of force is the Newton:

1 N = 1 kg m / s 2.

In cgs units, it is the "dyne":

1 d = 1 g cm / s 2 = 10 - 5 N.

If we assume for the moment that the mass of an object is not changing (not necessarily true: consider a rocket!), we see that

F = m dV / dt

= m a,

and we come to our only definition of mass: mass is a physical "constant of proportionality" relating force and acceleration. That is, to accelerate an object, it is necessary to apply a force, and that force is proportional to both the required acceleration and the mass of the object.

While mass is certainly an independent quantity, it can only be measured in conjunction with force and acceleration. For instance, the weight of an object is not equal to its mass, but rather to the product of its mass and the acceleration it feels due to the gravitational attraction of the Earth (or whatever planet or satellite you happen to be on!):

w = m g,

where g (in SI units) is assumed on the Earth to be constant at 9.81 m / s 2. In order to measure mass in space, we must apply a known force to an object and measure its acceleration. Its mass is then the ratio of the two.

Just as force is the time derivative of momentum, we can also write its components as the spatial derivatives of a different quantity, the "potential energy" U:

Fx = dU / dx,

Fy = dU / dy.

Potential energy is "energy of position", as opposed to kinetic energy, the energy an object has due to its motion. Both are scalars. Note that potential energy also has SI units of Joules, so that when we divide out one power of meters using the derivative, we obtain units of Newtons for force, as we should. Notice that an arbitrary constant can be added to U without changing the resulting force. This means that the position chosen to correspond to zero potential energy is arbitrary (but we must agree on a single position for any given problem!). It is important to note also that potential energy is only a useful quantity when friction is not involved. We will see why in a moment.

As an example of a system where potential energy is a useful quantity, consider gravity. As usual, assume that we are near the surface of the Earth and that g is a constant. In this case,

f = m g,

and that means that the potential energy must be

U = m g y

in order to satisfy the relation between force and potential energy. Here we use small letters for f and g since we there is only one dimension to the problem: vectors are unnecessary here. So the potential energy of an object is increased when it it is lifted higher, and decreases when the object is lowered. This fits our common sense: a professor has more energy when thrown off a higher building. It also helps us to see why friction must be absent (or negligible, as here we have neglected air resistance). Friction causes an object to expend additinal energy to attain a certain position. But the extra energy is required whether moving toward or away from that position. Hence friction is incompatible with our notion of potential energy, since potential energy should depend only on the position of the object and not on how it came to be in that position.

It is important to keep in mind the two alternate "definitions" of force in terms of derivatives. As a time derivative of momentum, force is necessary to change an object's momentum. Alternatively, when an object's momentum is changed by contact with another object, it exerts a force on that object (remember this the next time you get hit with a ball!). As a spatial derivative of potential energy, any object which has potential energy feels a force exerted on it. Alternatively, force is required to change the potential energy of the object (remember this one the next time you climb a flight of stairs!).

There are two other dynamic variables which we will use. Pressure is force per unit area; its SI unit the Pascal:

1 P = 1 N / m 2.

We also will discuss power, the time derivative of energy:

P = dE / dt.

The use of P for both Pascals and Power in one paragraph is confusing (even without using P for pressure!), but we will make the distinction clear whenever both quantities are used in the same problem. Here, the energy E is the the sum of the kinetic and potential energy:

E = K + U.

Power then has dimensions of energy over time, or in SI units, Watts:

1 W = 1 J / s.

When the force is constant,

P = f v.

Here, we deal only with scalar forces and velocities. The extension to vector quantities is possible, but unnecessary for our purposes.

Problem

The next section is about friction and drag .



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1996, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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