Foundations

Definitions

The basic entities are particles (point mass with no internal structure), space, time, motion, and force.

Place (position in Euclidean space), time (duration), force (cause of motion and secondarily of internal stress), and mass (inertia) are basic terms, whose values are from experiment. Their conceptual clarification is to be parallel with the development of theory.

Velocity and acceleration are the rate of change of position (as a vector in Euclidean space), and rate of change of velocity.

Momentum (a vector) is the product of mass and velocity.

Comments

Newton states his laws in terms of ‘bodies’. We state them in terms of particles, for the dynamics of extended bodies can be built from the dynamics of particles (which Newton does).

Regular and bold font indicate scalars and vectors, respectively.

Symbols

The symbols for position, time, force, mass, velocity, momentum, and acceleration are r, t, F, m, v, p, and a.

Newton’s laws in modern form

It is important that, though they may be analyzed separately, the laws stand together.

A particle that has no force on it remains at rest or in constant velocity (since velocity is a vector, motion is in a straight line.

Note this law is a special case of the second. It is stated here because it is traditional to do so. Some writers assert that it is necessary, for it establishes the existence of inertial frames. However, so does the second law.

The rate of change of momentum of a particle is equal to the force applied to it. In symbols,

F = ma.

Some statements say ‘net or external force applied’ instead of just ‘force applied’; however, for a particle with no internals structure, the two are the same.

If, relative to a second frame, the acceleration is a', and a = a' + ar, where ar is relative acceleration then the law is—

F = ma = ma' + mar = ma' – FI, where FI = -mar, and then,

F + Fi = ma' and Fi is an ‘inertial force’.

This shows that frames in which force = mass x acceleration are special and called inertial. If one frame is inertial, all frames moving at constant velocity but without rotation relative to it are inertial. Other frames are called non-inertial, and the force equation must be modified for them.

It also shows that given the validity of the force equation in original form, there are inertial frames.

When two particles have an interaction the forces of action and reaction are equal and opposite in direction and lie on the straight line through the particles.

Note that the distinction between action and reaction is not fundamental; the choice of which is which is usually according to context or what is the object of focus. Rather than ‘action and reaction’ it would be better to say, ‘the force exerted on one particle by the other and the force exerted on the other by the one’.

Units

In the SI system, the units of length, time, and mass, are the metre (M), second (s), and kilogram (Kg). These are the base units for mechanics.

The units of velocity and acceleration are then M/s (or Ms-1) and (M/s)/s (or M/s2 or Ms-2). These are derived units—i.e., in terms of the base units.

The unit for momentum (mass times velocity) is KgM/s or KgMs-1.

There are generally two options to define units for force—via Newton’s law (which makes the units of force derived) or independently (e.g., as the weight of one kilogram where the acceleration due to gravity has a standard value).

In the SI system, units for force are derived—1 Newton (1 N) is the force that accelerates 1 Kilogram at a rate of 1 M/s2. From the second law, 1 N = 1Kg x 1 M/s2 = 1 Kg-M/s2, or,

1 N = 1 KgM/s2.

Conversions

Though we prefer the SI system, you will find other systems in use. The most common is the Imperial System. It is necessary to have some familiarity with this system and how to convert between Imperial and SI.

Today, the units for the Imperial system are defined in terms of SI units. The imperial system uses the foot (ft) or 0.3048 M, the second (s), and the pound (lb) or 0.453 592 37 kg. A force unit, the poundal, 1 lb-ft/s2, is not in common use

The common force unit in the Imperial system in engineering use defines a pound force (lbf) as the force on 1 lb mass where acceleration due to gravity is the standard value 9.806 65 m/s2 or 32.174 048 556 ft/s2. Then, a different mass unit, the slug is defined as the mass that will be accelerated at 1fts-2 under a force of 1lbf. So, 1slug = 32.174 049 lb.

Now,

1 ft = 0.3048 M, or

[0.3048 M / 1 ft] = 1 (no units), i.e.,

[0.3048 M / 1 ft] = 1;

Note that this and all the following conversions can be inverted, thus—

[1 ft / .3048 M] = 1, or,

[3.280 839 895 ft / M] = 1, or,

[3.280 840 ft / M] = 1.

Similarly, for velocity and acceleration,

[0.3048 M s-1 / 1 fts-1] = 1,

[0.3048 M s-2 / 1 fts-2] = 1.

For mass,

1 slug = 32.174 048 556 lb = 32.174 048 556 x 0.453 592 37 kg = 14.593 902 937 kg, or,

[14.593 902 937 kg / 1 slug] = 1, or,

[0.068 521 765 858 slug / 1 kg] = 1, or,

For force,

1 lbf = 1 slug ft s-2 = 14.593 902 937 Kg x .3048 M s-2 = 4.448 221 615 N 4.448 N, or,

[1 lbf / 4.448 N] = 1.

For work (force times distance), the units are 1 ft lbf and 1 J (Joule, i.e., 1 N M)—

1 ft lbf = 0.3048 M x 4.448 N = 1.356 N M = 1.356 J, or,

[1 ft lbf / 1.356 J] = 1.

If units of power (work / time) are the ft lbf/s, and the Watt (W) or J/s, the numerical for the conversion is as above,

[1 ft lbf s-1/ 1.356 W] = 1.

When the Imperial Horsepower (HP) is defined as 550 ft lbf/s, the conversion is—

[550 ft lbf s-1/ 550 x 1.356 W] = 1, or,

[1 HP / 745.69 W] = 1.

Let us collect the results in a table

Quantity	SI to Imperial	Imperial to SI
dimensionless unit factors, brackets omitted
Length	3.281 ft / 1 M	0.3048 M / 1 ft
Velocity	3.281fts-1 / 1 M s-1	0.3048 M s-1 / 1 fts-1
Acceleration	3.281fts-2 / 1 M s-2	0.3048 M s-2 / 1 fts-2
Mass	0.068 522 slug / 1 kg	14.593 9 kg / 1 slug
Force	0.22481 slug / 1 N	4.448 N / 1 lbf
Work	0.737 600 ft lbf /J	1.356 J / 1 ft lbf
Power	0.0013410 HP / 1 W	745.69 W / 1 HP

Table 1 Conversion factors for mechanics

Dimensional analysis

The units of force and the units of mass x acceleration are the same. In SI they are the Newton or KgM/s2. Any derivation from the fundamental law involves multiplication (or division or differentiation) of quantities by the same quantity and addition (or integration or subtraction) of like quantity. There all equations derived from the force equation involve quantities with the same units.

The unit of force is the KgMS-2. We can write this in terms of dimensions [l][m][t]-2, where the dimensions are as follows—l is length, m is mass, and t is time. Consider a derived equation—

A = B, from which [A] = [B].

If the dimensions of A are

[A] = [l]a [m]b [t]c,

and the dimensions of B are

[B] = [l]α [m]β [t]γ.

Then, since [A] = [B],

a = α, b = β, and c = γ.

This is useful in ‘dimensional analysis’.

For example, if the frequency, f (dimension [t]-1), of a pendulum is a product of powers of l, its length, g, acceleration due to gravity, and mass m, then

[t]-1 = [f] = [l]a [g]b [m]c = [l]a [lt-2]b [m]c,

now, equating powers of the first and last terms

0 = a + b, -1 = -2b, 0 = c,

from which,

a = ½, b = ½, and c = 0.

Therefore,

f = k (g/l)½ , where k is a constant.

From simple harmonic motion for the pendulum, we know k = (1/2π).

Dimensional analysis gives us useful information without appealing to the physics, but the information is incomplete.

Forces

There are four known fundamental forces. Gravity and electromagnetism are the main two used in engineering mechanics.

The gravitational force between two particles is attractive, of magnitude, Gmm'/r2, where G is the gravitation constant, m and m' are the masses of the particles, and r is the distance between them.

On or near the surface of the earth, the force of gravity on a particle of mass, m, is mg (downward).

The contact force between two objects is usually resolved into a force N, normal to the contact surfaces, and friction F, tangential to the surfaces. The coefficient of friction, μ, is a property of the surfaces, such that sliding will not occur if the needed friction F N; otherwise sliding occurs. Friction is a manifestation of electrostatic forces, as is viscosity which is studied in fluid mechanics.

Summary

Newton’s laws are

The inertia law, F = ma, for motion in an inertial frame, and F = ma' + mar for a non-inertial frame.

The interaction between two particles is an action and reaction pair, which are equal, opposite, and lie on the straight line joining the particles.

For conversions, you are referred to Table 1, page 4.