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## Classical Mechanics

_{1}.

These laws were formulated by the great mathematician and physicist Isaac Newton (1642-1727). Much of Newton’s thought was inspired by the work of an individual who died the same year Newton was born, Galileo Galilei (1564-1642).

1. An object at rest tends to stay at rest; an object in motion tends to stay in motion

2. The acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to the object’s mass

3. Forces between objects always exist in equal and opposite pairs

Newton’s first law may be thought of as the *law of inertia*, because it describes the property of inertia that all objects having mass exhibit: resistance to change in velocity.

Newton’s second law is the verbal equivalent of the force/mass/acceleration formula: *F = ma*

Newton’s third law describes how forces always exist in pairs between two objects. The rotating blades of a helicopter, for example, exert a downward force on the air (accelerating the air), but the air in turn exerts an upward force on the helicopter (suspending it in flight). These two forces are equal in magnitude but opposite in direction. Such is always the case when forces exist between objects.

Work is the expenditure of energy resulting from exerting a force over a parallel displacement (motion)_{2}:

*W = Fx*

Where,

*W* = Work, in joules (metric) or foot-pounds (British)

*F* = Force doing the work, in newtons (metric) or pounds (British)

* x *= Displacement over which the work was done, in meters (metric) or feet (British)

Potential energy is energy existing in a stored state, having the potential to do useful work. If we perform work in lifting a mass vertically against the pull of Earth’s gravity, we store potential energy which may later be released by allowing the mass to return to its previous altitude. The equation for potential energy in this case is just a special form of the work equation (W = Fx), where work is now expressed as potential energy (W = Ep), force is now expressed as a weight caused by gravity acting on a mass (F = mg), and displacement is now expressed as a height (x = h):

*W = Fx*

*Ep = mgh*

*Where,*

*E**p* = Potential energy in joules (metric) or foot-pounds (British)

*m* = Mass of object in kilograms (metric) or slugs (British)

*g* = Acceleration of gravity in meters per second squared (metric) or feet per second squared (British)

*h* = Height of lift in meters (metric) or feet (British)

Kinetic energy is energy in motion. The kinetic energy of a moving mass is equal to:

Where,

Ek = Potential energy in joules (metric) or foot-pounds (British)

m = Mass of object in kilograms (metric) or slugs (British)

v = Velocity of mass in meters per second (metric) or feet per second (British)

The Law of Energy Conservation is extremely useful in projectile mechanics problems, where we typically assume a projectile loses no energy and gains no energy in its flight. The velocity of a projectile, therefore, depends on its height above the ground, because the sum of potential and kinetic energies must remain constant:

Ep + Ek = constant

In free-fall problems, where the only source of energy for a projectile is its initial height, the initial potential energy must be equal to the final kinetic energy:

Ep (initial) = Ek (final)

We can see from this equation that mass cancels out of both sides, leaving us with this simpler form:

It also leads to the paradoxical conclusion that the mass of a free-falling object is irrelevant to its velocity. That is, both a heavy object and a light object in free fall will hit the ground with the same velocity, and fall for the same amount of time, if released from the same height under the influence of the same gravity_{3}.

Dimensional analysis confirms the common nature of energy whether in the form of potential, kinetic, or even mass (as described by Einstein’s equation). First, we will set these three energy equations next to each other for comparison of their variables:

In all three cases, the unit for energy is the same: kilogram-meter squared per second squared. This is the fundamental definition of a “joule” of energy, and it is the same result given by all three formulae.

Power is defined as the rate at which work is being done, or the rate at which energy is transferred. Mathematically expressed, power is the first time-derivative of work (W):

The metric unit of measurement for power is the watt, defined as one joule of work performed per second of time. The British unit of measurement for power is the horsepower, defined as 550 foot-pounds of work performed per second of time.

Although the term “power” is often colloquially used as a synonym for force or strength, it is in fact a very different concept. A “powerful” machine is not necessarily a machine capable of doing a great amount of work, but rather (more precisely) a great amount of work in a short amount of time. Even a “weak” machine is capable of doing a great amount of work given sufficient time to complete the task. The “power” of any machine is the measure of how rapidly it may perform work. Downloaded from

An interesting exercise in dimensional analysis for people familiar with Joule’s Law in electric circuits shows just how work and power relate. Power, as you may recall, is defined in electriccircuits as the product of voltage and current:

P = IV

Showing the common units of measurement for each of these variables:

[Watts] = [Amperes] × [Volts] or [W] = [A][V]

Now we will substitute more fundamental units of measurement to show how the units comprising “power” really do come from the units comprising “volts” and “amps”. We know for example that the unit of the “ampere” is really coulombs of charge flowing per second, and that the unit of the “volt” is really joules of energy (or joules of work) per coulomb of charge. Thus, we may make the unit substitutions and prove to ourselves that the “watt” is really joules of energy (or joules of work) per second of time:

In summary, voltage is a measure of how much potential energy is infused in every coulomb of charge in an electric circuit, and current is a measure of how quickly those charges flow through the circuit. Multiplying those two quantities tells us the rate at which energy is transferred by those moving charges in a circuit: the rate of charge flow multiplied by the energy value of each charge unit.

Many instruments make use of springs to translate force into motion, or visa-versa. The basic “Ohm’s Law” equation for a mechanical spring relating applied force to spring motion (displacement) is called Hooke’s Law_{4}:

*F = −kx*

Where,

F = Force generated by the spring in newtons (metric) or pounds (British)

k = Constant of elasticity, or “spring constant” in newtons per meter (metric) or pounds per foot (British)

x = Displacement of spring in meters (metric) or feet (British)

Hooke’s Law is a linear function, just like Ohm’s Law is a linear function: doubling the displacement (either tension or compression) doubles the spring’s force. At least this is how springs behave when they are displaced a small percentage of their total length. If you displace a spring more substantially, the spring material will become strained beyond its elastic limit and either yield (permanently deform) or fail (break).

The amount of potential energy stored in a tensed spring may be predicted using calculus. We know that potential energy stored in a spring is the same as the amount of work done on the spring, and work is equal to the product of force and displacement (assuming parallel lines of action for both):

Ep = Fx

Thus, the amount of work done on a spring is the force applied to the spring (F = kx) multiplied by the displacement (x). The problem is, the force applied to a spring varies with displacement and therefore is not constant as we compress or stretch the spring. Thus, in order to calculate the amount of potential energy stored in the spring (Ep = Fx), we must calculate the amount of energy stored over infinitesimal amounts of displacement (F dx, or kxdx) and then add those bits of energy up ( R ) to arrive at a total:

We may evaluate this integral using the power rule (x is raised to the power of 1 in the integrand):

Where,

Ep = Energy stored in the spring in joules (metric) or foot-pounds (British)

k = Constant of elasticity, or “spring constant” in newtons per meter (metric) or pounds per foot (British)

x = Displacement of spring in meters (metric) or feet (British)

E0 = The constant of integration, representing the amount of energy initially stored in the spring prior to our displacement of it

For example, if we take a very large spring with a constant k equal to 60 pounds per foot and displace it by 4 feet, we will store 480 foot-pounds of potential energy in that spring (i.e. we will do 480 foot-pounds of work on the spring).

Graphing the force-displacement function on a graph yields a straight line (as we would expect, because Hooke’s Law is a linear function). The area accumulated underneath this line from 0 feet to 4 feet represents the integration of that function over the interval of 0 to 4 feet, and thus the amount of potential energy stored in the spring:

0

Note how the geometric interpretation of the shaded area on the graph exactly equals the result predicted by the equation : the area of a triangle is one-half times the base times the height. One-half times 4 feet times 240 pounds is 480 foot-pounds.

Rotational motion may be quantified in terms directly analogous to linear motion, using different symbols and units.

The rotational equivalent of linear force (F) is torque (τ ). Linear force and rotational torque are both vector quantities, mathematically related to one another by the radial distance separating the force vector from the centerline of rotation. To illustrate with a string pulling on the circumference of a wheel:

This relationship may be expressed mathematically as a vector cross-product, where the vector directions are shown by the right-hand rule (the first vector is the direction of the index finger, the second vector is the direction of the middle finger, and the product vector ~τ is the direction of the thumb, with all three vectors perpendicular to each other):

The proper unit of measurement for torque is the product of the force unit and distance unit. In the metric system, this is customarily the Newton-meter (N-m). In the British system, this is customarily the foot-pound (ft-lb) or alternatively the pound-foot (lb-ft). Note that while these are the exact same units as those used to express work, they are not the same types of quantities. Torque is a vector cross-product, while work is a dot-product (). The cross-product of two vectors is always another vector_{5}, while the dot-product of two vectors is always a scalar (direction-less) quantity. Thus, torque always has a direction, whereas work or energy does not.

Every quantity of force and motion which may be expressed in linear form has a rotational equivalent. As we have seen, torque (τ ) is the rotational equivalent of force (F). The following table contrasts equivalent quantities for linear and rotational motion (all units are metric, shown in italic font):

Linear quantity, symbol, and unit |
Rotational quantity, symbol, and unit |

Force (F) N | Torque (τ ) N-m |

Linear displacement (x) m | Angular displacement (θ) radian |

Linear velocity (v) m/s |
Angular velocity (ω) rad/sec |

Linear acceleration (a) m/s2 |
Angular acceleration (α) rad/s2 |

Mass (m) kg |
Moment of Inertia (I) kg-m2 |

Familiar equations for linear motion have rotational equivalents as well. For example, Newton’s Second Law of motion states, “The acceleration of an object is directly proportional to the net force acting upon it and inversely proportional to the object’s mass.” We may modify this law for rotational motion by saying, “The angular acceleration of an object is directly proportional to the net torque acting upon it and inversely proportional to the object’s moment of inertia.” The mathematical expressions of both forms of Newton’s Second Law are as follows:

F = ma τ = Iα

The calculus-based relationships between displacement (x), velocity (v), and acceleration (a) find parallels in the world of angular motion as well. Consider the following formula pairs, linear motion on the left and angular motion on the right:

An object’s “moment of inertia” represents its angular inertia (opposition to changes in rotational velocity), and is proportional to the object’s mass and to the square of its radius. Two objects having the same mass will have different moments of inertia if there is a difference in the distribution of their mass relative to radius. Thus, a hollow tube will have a greater moment of inertia than a solid rod of equal mass, assuming an axis of rotation in the center of the tube/rod length:

This is why flywheels_{6} are designed to be as wide as possible, to maximize their moment of inertia with a minimum of total mass.

The formula describing the amount of work done by a torque acting over an angular displacement is remarkably similar to the formula describing the amount of work done by a force acting over a linear displacement:

The formula describing the amount of kinetic energy possessed by a spinning object is also similar to the formula describing the amount of energy possessed by a linearly-traveling object:

_{1}Relativistic physics deals with phenomena arising as objects travel near the velocity of light. Quantum physics deals with phenomena at the atomic level. Neither is germane to the vast majority of industrial instrument applications.

_{2}Technically, the best way to express work resulting from force and displacement is in the form of a vector dot- product: . The result of a dot product is always a scalar quantity (neither work nor energy possesses a direction, so it cannot be a vector), and the result is the same magnitude as a scalar product only if the two vectors are pointed in the same direction.

_{3}In practice, we usually see heavy objects fall faster than light objects due to the resistance of air. Energy losses due to air friction nullify our assumption of constant total energy during free-fall. Energy lost due to air friction never translates to velocity, and so the heavier object ends up hitting the ground faster (and sooner) because it had much more energy than the light object did to start.

_{4}Hooke’s Law may be written as F = kx without the negative sign, in which case the force (F) is the force applied on the spring from an external source. Here, the negative sign represents the spring’s reaction force to being displaced (the restoring force). A spring’s reaction force always opposes the direction of displacement: compress a spring, and it pushes back on you; stretch a spring, and it pulls back. A negative sign is the mathematically symbolic way of expressing the opposing direction of a vector.

_{5}Technically, it is a pseudovector, because it does not exhibit all the same properties of a true vector, but this is a mathematical abstraction far beyond the scope of this book!

_{6}A “flywheel” is a disk on a shaft, designed to maintain rotary motion in the absence of a motivating torque for the function of machines such as piston engines. The rotational kinetic energy stored by an engine’s flywheel is necessary to give the pistons energy to compress the gas prior to the power stroke, during the times the other pistons are not producing power.

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