Mechanics Isaac Newton 25 December March 1727, Julian calendar 4 January March 1727, Gregorian calendar Books 1687: Philosophae


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1 Mechanics Isaac Newton 25 December March 1727, Julian calendar 4 January March 1727, Gregorian calendar Books 1687: Philosophae Naturalis Principia Mathematica, or Mathematical Principles of Natural Philosophy The Principia is justly regarded as one of the most important works in the history of science. Newton s Laws of Motion (1) Law of Inertia: (Galileo) A body free of forces moves in a straight line with constant speed. (2) A body of mass m kilograms under a force F Newtons has an acceleration a meter/second/second in the direction of the force with F = m a. The arrow over F, or a, means that it is a vector, so its direction is important. For example, a force of 10 Newtons acting on a 2 kg body gives it an acceleration of 5 m/s 2. (3) To every action there is always an equal and opposite reaction: or the forces of two bodies on each other are always equal and are directed in opposite directions. In other words, whenever a first body exerts a force F 1 upon 2 upon a second body, the second body exerts a force F 2 upon 1 upon the first body, such that F 1 upon 2 and F 2 upon 1 are equal in magnitude and opposite in direction.
2 Conservation of Linear and Angular Momentum: A particle of mass m and velocity v has linear momentum p = m v. The law of inertia says that p is constant for a particle moving free of forces. The interesting thing is that a system of N particles has total momentum p total = m 1 v 1 + m 2 v 2 + m 3 v m N v N. As these particles move, or collide, exerting forces on each other, the total momentum is constant if there are no external forces acting on the system. An example of this is the recoil of a rifle in the figure to the left below M = mass of rifle m=mass of bullet M much greater than m Rifle and bullet at rest, with zero momentum m V M Final momentum = 0, so MV = mv v V much smaller than v The rifle and bullet initially have zero velocities. After firing the forward momentum mv of the bullet ie equal and opposite to the backwards momentum MV of the rifle, so MV = mv, or V = m M v, with V much smaller than v because M is much bigger than m. Why is it that when a ballerina pulls in her arms, or a diver tucks in his knees, they spin faster? Rotational motion and angular momentum are more complicated. In many simple cases a body rotating with angular velocity ω has angular momentum L = I ω where I is the moment of inertia of the body, which depends on the mass, size, and shape of the body. For example, a sphere of mass M and radius R has I = 2MR2 5, but a cylinder has I = MR2 2. A body spinning free of external
3 torques moves with L constant. Say that L initil = L final, therefore I initial ω initial = I final ω final, then ω final = I initial I final ω initial. The interesting thing is that if I final becomes smaller than I initial, such as when a ballerina pulls in her arms, or a diver tucks in his knees, ω final becomes bigger tham ω initial and so the body spins faster. Acceleration: Galileo was the first to understand acceleration. Acceleration is the rate of change of velocity. A car that starts from rest with a constant acceleration a = 10 mph/second reaches a velocity v = 60 mph in a time t = 6 s. That is, v = at. The distance covered is d = v average t = 1 2 at2. Galileo discovered that if air resistance is negligible, all bodies near the Earth fall with the same acceleration g = 9.8m/s/s, or g = 9.8 m/s 2, no matter what is the weight of the object. That is, starting from rest, after t seconds, the speed of the object is v = gt, and the distance it falls is d = 1 2 gt2. So, after t = 1 second the speed is v = 9.8m/s 2 1s = 9.8 m/s, and the distance traveled is d = (1/2) 9.8m/s 2 1s = 4.9 m. After t = 3 sec the speed is v = 9.8m/s 2 3s = 29.4 m/s, and the distance is d = (1/2) 9.8m/s 2 (3s) 2 = 44.1 m, and so on. Circular motion (Huygens): A body moving with speed v in a circle of radius r has an acceleration with magnitude a = v 2 /r directed toward the center of the circle, even if it is
4 going around with constant speed. The acceleration is due to the fact that the velocity vector v is changing direction as it goes in a circle. m v a R Newtonian Gravity: Gravity has long range but is the weakest of the four fundamental interactions. The weight of an object is the force of the Earth s gravity upon the object. Newton explained his law of gravity saying that any two bodies with masses of m 1 kg and m 2 kg separated by a distance of r meters attract each other. Law of Motion F = m a m a F m 1 m 2 Force of gravity F F r The magnitude of the force of attraction is F gravity = Gm 1m 2 r 2 where G = Newton m 2 /kg 2. With this force the Sun attracts the planets and keeps them in orbit, or the Earth attracts an apple and accelerates it with g. From this we explain the acceleration of gravity g, because a body of mass m with acceleration g must be under a force F = m g, but the Earth of mass M E attracts a body of mass m on the surface of the Earth, at a distance R E from the center
5 of the Earth, with a force of magnitude F = GM Em r 2, so F = mg = GM Em RE 2, or g = GM E RE 2. The constant G is very small, and Newton did not know its value, but he knew g. When Cavendish measured G in 1798, the experiment was called weighing the Earth, because then, with R E m, M E = gr2 E G = 9.8 m/s 2 ( m) Newton m 2 /kg kg. Work and Energy Energy is the ability to do work. Energy is conserved. There are many types of energy. The most important is energy of motion, or Kinetic energy. When a body moves a distance d in the same direction of a total force F that acts on it during the entire motion, we say that the force has done work W = F d, and the body acquires kinetic energy KE = 1 2 mv2, where m is the mass, and v is the speed of the body. The unit of work and energy is the Joule. In order to be able to do work one must have a supply of energy to pay for it. For example a force of 5 Newtons acting over a distance of 3 meters does 15 Joules of work. If there is there is no movement, there is no work. So, for example, the columns that are supporting the roof are not lifting the roof and therefore are not doing any work and do not need a supply of energy. A body of mass m = 2 kg moving with speed v = 10 meter/second, has a kinetic energy
6 of 50 Joules. A Joule is the same as a Watt second. So a 60 Watt bulb, running for 10 seconds, consumes 600 Joules of energy. The kilowatthour is also an energy unit equal to 3,600,000 Joules. In your electric bill you pay about 10 cents for a kilowatthour of electrical energy. Potential Energy In order for an agent, such as a person or a machine, to slowly lift (little kinetic energy) a body of mass m kilograms by a distance h it is necessary for the agent to exert an upward force to overcome the weight of the body, which is W eight = mg, with g = 9.8 m/sec/sec. Therefore the work needed to lift the body is W ork = F orce distance = mgh. For example if m = 5 kg, and h = 2 meters, then =W ork = 5kg 9.8m/sec/sec 2meters = 98 Joules. This work is not manifested as kinetic energy of the body, but rather as Gravitational Potential energy, which is stored in the gravitational field and can be recovered, such as by letting the body drop. Then that energy is converted to kinetic energy of the body. In general, potential energy is the energy a body can have by virtue of its position in a field of force, such as gravity, electrical forces or nuclear forces. For example, a body at a height h has a gravitational potential energy P E = mgh. As a body falls and gathers speed, potential energy is converted to kinetic energy. Chemical energy is mostly due to the potential energy of electrical forces in atoms. Heat and Energy The internal energy of a body is the sum of the kinetic and potential energies of the random motions of its atoms. Temperature is a measure of the internal energy of a body. Recall that the Fahrenheit temperature T F is related to Celsius (or
7 Centigrade) temperatute T C by the formula T F = (9/5)T C +32. A more important temperature scale is the absolute (or Kelvin ) temperature T A which is related to T C by T A = T C The average kinetic temperature of a molecule of a body is related to T A by the formula KE average = 1 2 mv2 = 3 2 kt A where k = Joule/(degree Kelvin). Heat is the flow energy by virtue of a temperature difference, that either increases or decreases the internal energy of a body. The Calorie is an energy unit used to measure heat. The food calorie, or diet calorie, also called Calorie, or kilocalorie=4,186 Joules. Conservation of Energy The important principle of the conservation of energy, which says that energy is neither created nor destroyed, but only transformed, was not formulated until it was understood that heat is a form of energy. For example a book sliding on a table against friction slows down and stops. What happened to its kinetic energy, which seems to have disappeared? Well, it was converted to internal energy of the table and the book, which are both a little warmer than before. This can be checked because we know that the heat needed to raise the temperature of a body of mass m from an initial temperature T i to a final temperature T f is Heat energy = c m (T F T i ), where c m = c h m is the heat capacity of the body, and c h is the specific heat capacity per kilogram. For example, for water c h = 1 Calorie/kilogram, or 4,186 Joules/kilogram, which is large compared to most materials.
8 POWER The unit of power is the Watt. Power is the rate of doing work, or consuming energy. That is, if work W (in Joules) is done (or energy consumed) during a time t (in seconds), the power P is the work divided by the time, or P = W t in units of 1 Joule second = Watt, or 1Joule = 1Watt second For example a 60 Watt lightbulb running for 3 seconds consumes W = 60 Watt 3 seconds = 180 Joules. The four Fundamental Interactions Gravity: It acts upon massenergy, which is conserved. It is the weakest, and long range. It is an attractive force. It keeps planets, stars, galaxies together. general relativity explains it in terms of a curvature of spacetime. Quantum gravity is not yet a successful theory. The Weak Interaction: It acts upon the three flavors of lepton number. The electron, muon, and τ lepton flavor numbers, are separately conserved. The electron, muon, and the τ, as well as their associated electron neutrino, muon neutrino, and τ neutrino have lepton number +1 (each of its kind). Their antiparticles have 1 lepton numbers of each kind. It is weak and very short range (nuclear dimensions). It is important in radioactive decay, for example, neutron proton + electron + electron antineutrino. It is mediated by an exchange of W +, W, and Z 0 mesons. Electromagnetic Interaction: It acts upon electric charge, which is conserved. The proton has charge +e = Coulombs and the electron has e. Opposite charges repel
9 and like charges attract each other. It is strong and long range enough that we have built a great technology using it. It keeps atoms together. It is mediated by an exchange of photons. The Strong Interaction: It acts upon color charge: red, green, and blue. Color charge is conserved. Color charge is carried by quarks, which come in 6 flavors, u, d, s, c, t, and b. A proton is made of uud, and a neutron is made of ddu. It is the strongest and short range (nuclear dimensions). It keeps the nucleus together, so the positive protons will not fly apart. It is mediated by the exchange of colored gluons, which have a color and anticolor, like red and antigreen.
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