Newton's First Law

Newton's First Law of Motion states that in order for the motion of an object to change, a force must act upon it, a concept generally called inertia.

About Physical Laws: About Physical Laws:

Over the years, one thing scientists have discovered is that nature is generally more complex than we give it credit for. The following laws of physics are considered fundamental, but many of them refer to idealized, closed systems, which are hard to obtain in the real world. 

Faraday's Law of Induction

In physics, a quantitative relationship between a changing magnetic field and the electric field created by the change, developed on the basis of experimental observations made in 1831 by the English scientist Michael Faraday.

Faraday discovered that, whenever the magnetic field about an electromagnet was made to grow and collapse by closing and opening the electric circuit of which it was a part, an electric current could be detected in a separate conductor nearby. Moving a permanent magnet into and out of a coil of wire also induced a current in the wire while the magnet was in motion. Moving a conductor near a stationary permanent magnet caused a current to flow in the wire, too, as long as it was moving.

Faraday visualized a magnetic field as composed of many lines of induction, along which a small magnetic compass would point. The aggregate of the lines intersecting a given area is called the magnetic flux. The electrical effects were thus attributed by Faraday to a changing magnetic flux.

Years later the Scottish physicist James Clerk Maxwell proposed that the fundamental effect of changing magnetic flux was the production of an electric field, not only in a conductor but also in space even in the absence of electric charges. Maxwell formulated the mathematical expression relating the change in magnetic flux to the induced electromotive force (E, or emf).

This relationship, known as Faraday's law of induction, states that the magnitude of the emf induced in a circuit is proportional to the rate of change of the magnetic flux that cuts across the circuit.

If the rate of change of magnetic flux is expressed in units of webers per second, the induced emf has units of volts.

Generalized Hooke's Law

The generalized Hooke's Law can be used to predict the deformations caused in a given material by an arbitrary combination of stresses.

The linear relationship between stress and strain applies for   

where:

E is the Young's Modulus n is the Poisson Ratio

The generalized Hooke's Law also reveals that strain can exist without stress. For example, if the member is experiencing a load in the y-direction (which in turn causes a stress in the y-direction), the Hooke's Law shows that strain in the x-direction does not equal to zero. This is because as material is being pulled outward by the y-plane, the material in the x-plane moves inward to fill in the space once occupied, just like an elastic band becomes thinner as you try to pull it apart. In this situation, the x-plane does not have any external force acting on them but they experience a change in length. Therefore, it is valid to say that strain exist without stress in the x-plane.

Ideal Gas Law

An ideal gas is defined as one in which all collisions between atoms or molecules are perfectly elastic and in which there are no intermolecular attractive forces.
Ideal gas law is a generalization containing both Boyle's law and Charles's law as special cases and states that:
In such a gas, all the internal energy is in the form of kinetic energy and any change in internal energy is accompanied by a change in temperature. An ideal gas can be characterized by three state variables:
·         absolute pressure (P), ·         volume (V), ·         and absolute temperature (T).
The relationship between them may be deduced from kinetic theory and is called the Ideal gas law.
PV = kT = nRT
where
·         n is the total number of moles, ·         NA = Avogadro's Number = 6.02217 · 1023 molecules/mole, ·         R = Universal gas constant = 8.314 J/K · mol , ·         k = Boltzmann Constant = R/NA = 1.380622 · 10-23 J/K.
The ideal gas law can be viewed as arising from the kinetic pressure of gas molecules colliding with the walls of a container in accordance with Newton's laws. But there is also a statistical element in the determination of the average kinetic energy of those molecules. The temperature is taken to be proportional to this average kinetic energy; this invokes the idea of kinetic temperature.

Kepler's Laws

The German astronomer Johannes Kepler, who was Brahe's assistant, acquired Brahe's astronomical data and spent about 16 years trying to deduce a mathematical model for the motion of the planets. After many laborious calculations, he found that Brahe's precise data on the resolution of Mars about the Sun provided the answer. Such data are difficult to sort out because the Earth is also in motion about the Sun.

Kepler's analysis first showed that the concept of circular orbits about the Sun had to be abandoned. He eventually discovered that the orbit of Mars could be accurately described by an ellipse with the Sun at one focal point. He then generalized this analysis to include the motion of all planets. The complete analysis is summarized in three statements, known as Kepler's laws:

1.	All planets move in elliptical orbits with the Sun at one of the focal points.
2.	The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals.
3.	The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit.

Half century later, Newton demonstrated that these laws are the consequence of a simple force that exists between any two masses. Newton's law of gravity, together with his development of the laws of motion, provides the basis for a full mathematical solution to the motion of planets and satellites. More important, Newton's law of gravity correctly describes the gravitational attractive force between any two masses.

Mathematical statements:

Kepler's second law

Where dA is the area swept by radius vector r in a time dt and Mp is the planet mass.

Kepler's third law

Where KS is a constant given by

MS is the Sun mass, G is universal gravitational constant and T is the time.

Law of Atmospheres and Boltzmann Law

The law of atmospheres, also known as the barometric law, states that the pressure n(y) as a function of height y varies as:

According to the ideal gas law, a gas of N particles in the thermal equilibrium obeys the relationship PV = NkBT. It is convenient to rewrite this equation in terms of the number of particles per unit volume of gas, nV = N/V. This quantity is important because it can vary from one point to another. In fact, our goal is to determine how nV changes in our atmosphere. We can express the ideal gas law in terms of nV as P = nVkBT. Thus, if the number density nV is known, we can find the pressure and vice versa.

The pressure in the atmosphere decreases as the altitude increases because a given layer of air has to support the weight of the air above it — the greater the altitude, the less the weight of the air above that layer and the lower the pressure.

To determine the variation in pressure with altitude, consider an atmospheric layer of thickness dy and the cross-sectional area A. Because the air is in static equilibrium, the upward force on the bottom of this layer, PA, must exceed the downward force on the top of the layer, (P + dP)A, by an amount equal to the weight of gas in this thin layer. If the mass of gas molecule in the layer is m, and the area a total of N molecules in the layer, then the weight of the layer is  w = mgN = mgnVAdy. Thus A - (P + dP)A = mgnVAdy, Which reduces to dP = - mgnVdy Because P = nVkBT, and T is assumed to remain constant, therefore dP = nVkBT dnV.

Substituting this into the above expression for dP and rearranging gives

Integrating this expression, we find

Boltzmann distribution law

Boltzmann distribution law is important in describing the statistical mechanics of a large number of particles. It states that the probability of finding the particles in a particular energy state varies exponentially as the negative of the energy divided by kBT. All the particles would fall into the lowest energy level, except that the thermal energy kBT tends to excite the particles to higher energy levels.

Distribution of particles in space is

Where n0 is the number of particles where U = 0 This king of distribution applies to any energy the particles have, such as kinetic energy. In general, the relative number of particles having energy E is

Ohm's Law

Ohm's law was named after George Simon Ohm (1787 — 1854). It states that for many materials (including most metals), the ratio of the current density to the electric field is a constant,  that is independent of the electric field producing the current. A courant density  and an electric field E are established in a conductor when potential difference is maintained across the conductor. If the potential difference is constant, the current is also constant. Very often, the current density is proportional to the electric field:

Where the constant of proportionality  is called conductivity of the conductor.

A form of Ohm's law useful in practical applications can be obtained by considering a segment of a straight wire of cross-sectional area A and length l.  A protential difference V = Vb - Va is maintained across the wire, creating an electric field in a wire and current.  If the electric field in the wire is assumed to be uniform, the potential difference is related to the electric field through the relationship

A form Therefore, the magnitude of the current density in the wire can be expressed as

Since, the potential diffence can be written

The quantity  l A is called the resistance R of the conductor.  It is defined as the ratio of the potential difference across the conductor to the current

This relationship is referred to as the basic law in electric circuits, the Ohm's law. The SI unit of resistance (R), which is called the ohm , is in fact a derived unit, related to the voltage and current units through

Pascal's Law

Also called Pascal's Principle

Pascal's law — developed by French mathematician Blaise Pascal — states that when there is an increase in pressure at any point in a confined fluid, there is an equal increase at every other point in the container.

Definition of pressure: If F is the magnitude of the normal force on the piston and A is the surface area of a piston, then the pressure, P, of the fluid at the level to which the device has been submerged as the ratio of the force to area.

Since the pressure is force per unit area, it has units of N/m2 in the SI system. Another name for the SI unit of pressure is Pascal (Pa)

An important application of Pascal's law is the hydraulic press. A force F1 is applied to a small piston of area A1. The pressure is transmitted through a liquid to a larger piston of area A2. Since the pressure is the same on both sides, we see that P = F1/A1 = F2/A2. Therefore, the force F2 is larger than F1 by multiplying factor A2/A1. Hydraulic brakes, car lifts, hydraulic jacks, and forklifts all make use of this principle.

Right-Hand Rule

There are a few forms of this rule, and it can be applied in many ways. Originally, it was a trick for right-handed coordinate systems to determine the direction of the magnetic field surrounding a long, straight wire carrying a current. Note that the magnetic field lines form circles around the wire.

This is how it works. Orient your right hand so that your thumb is along the direction of the current. The four fingers wrap in the direction of the magnetic field.

This has immediate application for determining the orientation of the z-axis basis unit vector, , in terms of the x- and y-axes' basis unit vectors; curl your right hand in the direction of to , and your thumb will point in the direction of cross . The rule is also applicable in several practical applications, such as determining which way to turn a screw, etc. There is also a left-hand rule, which exhibits opposite chirality.

The right hand rule applies to the flows of positive charges. If negative changes are flowing, simply use your left hand. The right hand rule can also be used to find the force induced by the presence of a magnetic field and the velocity of a particular particle. When a charge is placed in a magnetic field, that charge experiences force if:

1. the charge is moving relative to the magnetic field,
2. the charge's velocity has a component perpendicular to the direction of the magnetic field

The Biot-Savart Law

Shortly after Oersted's discovery in 1819 that a compass needle is deflected by a current-carrying conductor, Jean Baptist Biot and Felix Savart reported that a conductor carrying a steady current exerts a force on a magnet.

From their experimental results, Biot and Savart arrived in an expression that gives the magnetic field at some point in space in terms of the current that produces the field. The Biot-Savart law says that if a wire carries a steady current I, the magnetic field dB at a point P associated with an element of the wire ds has the following properties:

The Laws of Motion

These force laws, together with the laws of motion, are the foundations of classical mechanics. They are based on experimental observations and were formulated more than three centuries ago by Isaac Newton(1642-1727).

1st Law of Motion:

An object at rest remains at rest and an object in motion will continue in motion with a constant velocity (that is, constant speed in a straight line) unless it experiences a net external force.

In other words, when the net force on a body is zero, its acceleration is zero. That is, when , then a = 0.

Where F is the force on a body and a is its acceleration.

Newton's first law is sometimes termed simply the "Law of Inertia".

2nd Law of Motion:

Newton stated that the force on a particle is equal to the rate of change of its linear momentum, which is the product of its mass and velocity. In other words, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Mathematical statement of Newton's second law:

Vector expression:

Component equations

Units of Force and Mass The SI unit of force is the newton, which is defined as the force that, when acting on a 1-kg mass, produces an acceleration of 1 m/s2. From this definition and Newton's second law, we see that the newton can be expressed in terms of the following fundamental units of mass, length, and time:

Definition of newton: 1 N = 1 kg · m/s2

The unit of force in cgs system is called the dyne and is defined as the force that, when acting on a 1-g mass, produces an acceleration of 1 cm/s2:

Definition of dyne: 1 dyne = 1 g · cm/s2

In the British engineering system, the unit of force is the pound, defined as the force that, when acting on a 1-slug mass, produces an acceleration of 1 ft/s2:

Definition of pound: 1 lb = 1 slug.ft/s2 Since 1 kg = 103 g and 1 m = 102 cm, it follows that 1 N = 105 dynes. It is left as a problem to show that 1 N = 0.225 lb. The slug is the unit mass in the British engineering system and is that system's counterpart of the SI kilogram.

Units of force, Mass, and Acceleration

System of Units Mass Acceleration Force SI kg m/s2 N = kg · m/s2 cgs  g cm/s2 dyne = g · cm/s2 British engineering slug ft/s2 lb = slug.ft/s2

3rd Law of Motion:

Newton's third law states that if two bodies interact, the force exerted on body 1 by the body 2 is equal to and opposite the force exerted on the body 2 by body 1.

F12 = - F21

In other words, forces always occur in pairs of that a single isolated force cannot exist. The body 1 exerts on body 2 is sometimes called action force; while the force body 2 exerts on body 1 is called the reaction force. In reality, either force can be labeled the action or reaction force. The action force is equal in magnitude to the reaction force and opposite in direction. In all cases, the action and reaction forces act on different objects.

Thermodynamics

Thermodynamics is the study of relationship between energy and entropy, which deals with heat and work. It is a set of theories that correlate macroscopic properties that we can measure (such as temperature, volume, and pressure) to energy and its capability to deliver work. A thermodynamic system is defined as a quantity of matter of fixed mass and identity. Everything external to the system is the surroundings and the system is separated from the surroundings by boundaries. Some thermodynamics applications include the design of:

air conditioners and refrigerators turbo chargers and superchargers in automobile engines steam turbines in power generation plants jet engines used in aircraft

Zeroth Law of Thermodynamics

The zeroth law of thermodynamics states that when two bodies have equality of temperature with a third body, they in turn have equality of temperature with each other. All three bodies share a common property, which is the temperature. For example: one block of copper is brought into contact with a thermometer until equality of temperature is established, and is then removed. A second block of copper is brought into contact with the same thermometer. If there is no change in the mercury level of the thermometer during this process, it can be said that both blocks are in thermal equilibrium with the given thermometer.

First Law of Thermodynamics

The first law of thermodynamics states that, as a system undergoes a change of state, energy may cross the boundary as either heat or work, and each may be positive or negative. The net change in the energy of the system will be equal to the net energy that crosses the boundary of the system, which may change in the form of internal energy, kinetic energy, or potential energy. The first law of thermodynamics can be summarized in the equation:

Where:  is the heat transferred to the system during the process   is the change in internal energy  is the change in kinetic energy  is the change in potential energy  is the work done by the system during the process