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- Published: Monday, 14 August 2023 22:58
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Force:
There are 4 fundamental forces or interactions in nature. All the motion that you see around us are due to forces. These forces may be friction, gravitation, push, etc. They ultimately belong to one of these 4 forces. It's easier to visualize these forces by considering a field, and then this field exerting forces. Force is always a vector as the effective force due to multiple forces is a vector sum of these forces. We are omitting the arrow sign on top of force for simplicity, but it should always be there to indicate it's a vector. The 4 fundamental forces or fields are:
- Gravitational force: This is attraction force due to mass of an object, and it's range is infinite. Newton's formula for gravitational force (published in 1686) gives us the formula for this:
- Newton's Gravitational Law: Force between 2 bodies of mass m1 and m2 as F = G*m1*m2/R^2, where G=Gravitational Constant of 6.674*10^-11 m^3/(kg*s^2). Here both F and R are vectors.
- Gravitation Field is defined as field due to presence of mass M and is given as g = F/m = G*M/R^2. The other mass is removed in eqn here, as F =m*g where g is the Gravitation Field due to mass M,
- The value of G wasn't determined at Newton's time. It was used to calculate forces relative to one another. It was in 1798 (>100 years later) that value of G was found out experimentally. It can be seen that this gravitational force is very weak (as G is a very small number), and is only relevant when at least one of the bodies is very large as moons, planets, stars, etc. What in the mass causes gravity or gravitational field is unknown. Even Newton hypothesized that cause of this power is unknown and contrary to sound science.
- Electromagnetic Force: This is the attractive or repulsive force due to charge. Look under "ElectroMagnetism" section.
Newtonian Laws:
3 of Newton's Laws shaped human advancement, as his 3 laws were revolutionary and considered one of the greatest scientific advancements. His book "Philosophiæ Naturalis Principia Mathematica", published in 1987 in Latin, contained his 3 laws. An excerpt of his laws from the original Latin and translated to English is here => https://www.gsjournal.net/Science-Journals/Communications-Mechanics%20/%20Electrodynamics/Download/4537
Another link for the 3 Newronion Laws and their translation => https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/66%3A_Appendices/66.18%3A__Newtons_Laws_of_Motion_(Original)
Below are Newton's 3 laws, directly translated from his original book.
- First law: Law of Inertia => Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.
- Second law: Force and momentum => The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. We write it as F=m*a, but in reality it's F=d/dt(m*v) or Force is equal to the rate of change of momentum.
- Third law: Action and Reaction: To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. Most important thing to note is that the 2 forces of action and reaction are acting on 2 different bodies.
Ex of force/work => To get an idea of how much 1N of force is, consider a 100g weight that you carry on your hand. That is 1N of force that you are applying to counteract earth's gravitational force. Now if you lift it up by 1m, that's 1J of work that you did. So, 1N of force and 1J of force look like very small amounts, as your body doesn't feel any tiredness doing this.
Pressure: Pressure is Force per unit area. i.e P = F/A. SI unit of P is Pascal (Pa) which is 1N/m^2. 1 N is the force equiv to holding 100g of weight. This doesn't sound like too much weight. Now if we spread this weight over 1m^2, that's almost negligible Pressure. 100g of weight over 100cm*100cm = 0.01g/cm^2 of weight is 1 Pa. It's more convenient to use 1kPa (1 kilo pascal) or 1 bar (=100 kPa). 1kPa is 10g/cm^2, while 1 bar is 1kg/cm^2. Atmospheric pressure due to air is measured in atm (atmospheric pressure), where pressure at sea level is defined as 1 atm=101,325 Pa, which is ~1 bar (To be precise, 1 atm is slightly greater than 1 bar). Pressure is also measured in inches/mm of Mercury (Hg) where pressure at sea level = 760mm = 29.9 inches (approximated to 30 inches) = 1 atm pressure.
In USA, pressure is measured in pounds of force per square inch (psi)—for example, in car tires. 1 lb=453.6g => 1 lb force = 453.6g*9.8N/kg = 4.445N. So 1 psi = 4.45N/(0.0254)^2=6897.5N/m^2 = 6.9kPa. => 1 bar = 100/6.9=14.5 psi. So, 1 atm=14.7 psi. Pressure readings in excess of atmospheric pressure are usually more relevant in measurements, so we use psig (pounds per square inch gauge) in such cases. PSIG is measured relative to atm pressure, i.e 1 PSIG = 1 PSI - 14.7, so at sea level PSIG =0. PSI measures pressure relative to vacuum.
Barometer: Barometers are used to measure pressure. We take a tube closed at end end, and fill it with liquid as Mercury or water. Now, if we invert and immerse it in container of same liquid, then liquid inside the tube will rise until the point where pressure due to weight of liquid (+ any air/vapor pressure inside the tube in the empty space above the liquid) inside is same as pressure due to air outside. Mercury rises to 760 mm (or 2.5 ft), while water rises to 34 ft (since Hg is ~13,6 times denser than water). P = h*ρ*g.
In pressure gauges (usually in HVAC applications), pressure is usually shown as PSIG. In a barometer, if the air inside the tube is at atm pressure, then the mercury level will be at 0 inside the tube. But if the air inside the tube is in perfect vacuum, then the mercury will rise to 30 inches. We take these readings inverted to indicate relative pressure and indicate it as inch Hg Pressure, So a pressure reading of "0 in Hg" means the pressure is same as atmospheric pressure, while a "-30 in Hg" pressure means absolute 0 pressure. These can also be shown in bar as "0 bar" for 1 atm pressure and "-1 bar" for absolute 0 pressure. However, showing these on linear scale is hard, because we can never go to 0 vacuum. We keep on going down by a factor of 10 or something infinitely. So, a log scale is preferred in many gauges, where very precise measurement is required.
Micron unit: Since pressure becomes very low close to vacuum, we use another unit called micron in pressure gauges to indicate very low level of pressures. 1 mm Hg of pressure is divided into 1000 parts called microns. So, 1 micron pressure = 0.001 mm of Hg pressure. At 1kPa (or 0.01 bar), water boils at 10C. We usually try to get to this vacuum level, when we want to get rid of moisture in any system (as at this pressure, liquid water will get converted to steam, and get sucked out via vacuum pump). 100 kPa = 760 mm of Hg, so 1kPa=7.6mm Hg or 7600 micron. As a thumb rule, going to 1% of atm pressure starts the process of water to boil at room temp. People in HVAC industry accept a pressure of 1000 micron or lower in order to guarantee that no water is present in the evacuated system. Usually 500 micros or lower is what people shoot for when vacuuming an HVAC system. These kind of low pressures can only be read by "micron gauge", which are very expensive ($200 for branded ones)
1 Pascal = 7 micron of Hg (implying 1 micron is extremely low pressure, since 1 Pascal is already too low of pressure)
1000 micron = 1 mm of Hg pressure = 0.02 psi
100K micron = 100 mm of Hg pressure = 2 psi or 0.1 atm (this is the highest pressure that a micron gauge will read, since it's designed to read vacuum pressure and NOT high pressure)
Manometer: These are similar to Barometer, and are used to measure pressure of gases. Manometers are used in measuring blood pressure using device called a sphygmomanometer (Greek sphygmos = “pulse”).
Work and Energy:
At the time of Newton, concept of Work or energy wasn't known. The concept of work was introduced by the French mathematician Gaspard Coriolis in 1826. Later, James Prescott Joule did experiments where he would have a calorimeter, with paddles inside so that moving the paddles would churn the water inside, and the friction of that churning would raise the temperature of the water. One of the ways of driving the paddles would be to have a weight suspended on a string, and the string running over a pulley so that the weight descending would drive the paddles.
As we know: the outcome of those experiments was that the temperature rise of the water is proportional to the height over which the weight descended. Also, Joule verified that the amount of energy transfer is independent of how fast or how slow the weight descends. The weight can drop fast or slow, at the end the same amount of energy is transferred.
So, work is only done when a force is able to move the body a certain distance in the direction of force. If the body doesn't move, then no work will be done (temp of water won't rise with weight just hanging there, and not moving).
Work: Mathematically, the work done by an individual force on an object during a displacement is defined as dot product of Force and displacement (i.e displacement in the direction of force)
W(t) = F(t) . r(t). Work is a scalar and has units of Joules (1J = 1N-m = 1kg-m/s^2 *m = kg-(m/s)^2)
ΔW(s) = F(s) . Δs, where the incremental work done by Force is incremental displacement done by that force (in the direction it's applied, hence dot product). To find total work done, it's integration of func or area under curve as W = ∫ F(s).ds (NOTE: -ve are under curve means negative work done in that portion)
It can also be written in terms of time "t", as W = ∫ F(t) . ds/dt.dt = ∫ F(t).v(t)dt. Here Force is rep as func of time "t", while in above eqn, Force was rep as func of displacement.
Force does +ve work when it acts in the same dir as the displacement of the body. It does -ve work, when it's in opposite dir. If it is perpendicular to dir of motion (as in centripetal force), it does no work, and that's why the velocity of body in circular motion remains constant.
Power: It's Work done per unit time = P(t) = dW/dt = F(t).dx/dt = F . v (dot product of Force and velocity at that instant in time)
Energy: Energy can be defined as the measurement of the ability of something to do work. When work is being done by a force on some body, then that work done gets transferred as energy to that body. The body which applied the force to do work loses the exact amount of energy. Now this energy can be in any form as heat, light, motion, etc. In kinematics, we are mostly dealing with motion, so we'll consider only Kinetic energy (KE) which is energy in a body due to motion (speed). Energy is a property of the system. For most practical usage, work and energy are used interchangeably as they define the same thing, and have the same units of Joules.
Work-Kinetic Energy (or Work-Energy (WE)) theorem: The net work done on an object during a displacement is equal to the change in the object's kinetic energy during that displacement (assuming a rigid body, so that there are no internal degrees of freedom, and hence vibrational energy are 0)
∑Wi = ΔK, where summation refers to sum of all work (net work) done on the body in dir of motion, and ΔK is the change in Kinetic energy of the rigid body.
=> ||∑F||.||d|| cos(Θ) = ΔK, where ∑F finds the net force, and then multiply it with distance moved, where Θ is the angle b/w the net force and the displacement of the body.
Potential Energy (PE): This is a new kind of energy defined for conservative forces.
Conservative force: It's a force with the property that the total work done by the force in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work done (the sum of the force acting along the path multiplied by the displacement) by a conservative force is zero. Ex of conservative forces are spring force, gravitational force, electric force, magnetic force, etc. Ex of non conservative force is friction force, as in a closed loop path, work done by friction force is never 0, as force is always opposite to dir of motion, no matter how the body moves.
Central forces: These forces act along the line joining the centres of two bodies. A central force is conservative if and only if it is spherically symmetric. Ex of central forces are electric force, magnetic force, gravitational force, etc.
Another property that I suppose is true for conservative forces, is that the direction of force is constant, irrespective of the direction of motion. So, irrespective of what path the body takes, the dot product is always the same. Hence, path doesn't matter for such forces.
For conservative forces, we may define another Energy, called Potential Energy (PE), which is the energy stored which has the "potential" to do work. This energy usually makes it easier to work in presence of conservative forces, as field of a conservative force has ability to do work. Imagine, a large body having gravitational field (direction of force/field is pointing inwards towards the body) around it. It will attract any other body which is in it's field. If other body is at infinity, then field can pull it towards itself inc it's speed for ever until it collides. Think of it, as stored PE of the 2 body system, and when released, it allowed that PE to do work and be converted to KE. Now, if we start the other body from somewhere close to the large body, then there is less field available to do work, so speed will increase, but now it will collide at a lower speed, as it has less KE (due to less field available to do work).
We define PE of any system as -ve of work done by this field. Since W = ∫ F.dx = ΔKE, Hence ΔPE = - ∫ F.dx = -ΔKE => ΔPE + ΔKE = 0 => PE + KE = constant, in presence of conservative force. This is known as law of "conservation of Energy (CoE)".
Previously, we used WE theorem (by not defining PE), but now, we can use conservation of energy by defining PE for conservative forces. As long as the only forces present are conservative forces, we should use CoE, as it's easier to solve it this way, instead of using WE thm, which will also give same result (by not using concept of PE at all).
PE of Gravitational field:
To find PE due to Gravitational field, we use ΔPE = - ∫ F.dx => PE(x2) - PE(x1) = - ∫ GM.m/r^2.dr (lower bound=x1 to upper bound=x2) =