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Physics for Scientists and Engineers

by: Marjorie Hahn

Physics for Scientists and Engineers PHYSICS H7A

Marketplace > University of California - Berkeley > Physics 2 > PHYSICS H7A > Physics for Scientists and Engineers
Marjorie Hahn

GPA 3.95


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Class Notes
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This 9 page Class Notes was uploaded by Marjorie Hahn on Thursday October 22, 2015. The Class Notes belongs to PHYSICS H7A at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 50 views. For similar materials see /class/226695/physics-h7a-university-of-california-berkeley in Physics 2 at University of California - Berkeley.

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Date Created: 10/22/15
Physics H7A Fall 2003 D Budker Plan of Lectures Lecture one August 26 Introduction names of Instructor and GSI contact information web page class e mail list of ce hours labs discussion sections midterms The goal of the course 7 professional introduction to mechanics as the basis of all further physics 0 The role ofmath ijust a tool the ideas will be physical not mathematical dimensions limiting cases orders of magpitude vs formal mathematical reasoning example with a student her boyfriend and ice cream odd number vs too expensive 0 Systems of units defined basically by units of length mass and time Two most commonly used systems are SI MKS and Gaussian CGS Many working physicists still use CGS as it is particular convenient for EampM In mechanics it really does not matter and we will use all kinds of units Warning watch out for unit consistency Use of the KampK book We will heavily rely on the book I find it silly to repeat everything that is so well written there in class So in general we will be going over examples that are not necessarily in the book while I will be assuming that you have read the chapters that I will assign For example this week please read Chapter one on mathematical preliminaries Homework will include a mixture of origina problems and those from KampK It is quite essential to do at least halfa dozen problems a week in order to keep up with the course Enough of preliminaries time to get to business Scalars vs Vectors Scalars are quantities that do not have any spatial direction associated with them time mass number of items distance while vectors are directed line segments with which we associate both a direction generally in 3d Cartesian space but sometimes restricted to lower dimensions eg 2 in the case of a plane or 1 for linear problems and a scalar representing the length of the vector There are many notations people use for vectors for example bold letters like V and letters with arrows on top of them likex7 In the special case of vectors with unit length a common notation is a hat on top of the letter 51 Properties of vectors Vectors can be added subtracted multiplied or divided by scalar you can use parenthesis permute things pretty much in the usual way An example addition of two vectors httpsocratesberkeleyedubudkerPhysicsH7A l Physics H7A Fall 2003 D Budker Ql Vi E b Ql gt b Now you can also multiply two vectors and there are two very different way of doing it Scalar Product 5E 5H5 cos6 which is a scalar and where vertical bars designate the length of the vector and 6 is the angle between the vectors and Vector Product 5 X E which is a vector that is directed according to the right hand rule and whose length is JHE sinH We will discuss this in some detail Incidentally there is generally another thing one can make out of two vectors called a tensor which is neither a scalar nor a vector but we will not deal with tensors for now Cartesian coordinates El axf ayfz 612 Scalar and vector product expressed We have already talked a bit about the difference in the way mathematicians and physicists think and about the importance of dimensions Turns C out that you can often say a lot about a problem just from knowing the units in which the answer a should be measured Here is an example due to a great theoretical physicist AB Migdal which is frowned upon by formal mathematicians We will prove the Pythagoras Theorem by the method of I dimensions The Theorem states that in a right b in the coordinate notation https0cratesberkeleyedubudkerPhysicsH7A 2 Physics H7A Fall 2003 D Budker angle triangle the sum of the squares of the side lengths is equal to the square of the length of the hypotenuse a2 b2 c2 To prove this let s drop a perpendicular from the rightangle comer onto the hypotenuse as shown Clearly the area of the original triangle is the sum of the areas of triangles I and II Now we know that a rightangle triangle is fully de ned by two of its elements for example the length of the hypotenuse and one ofthe angles say B Since area is measured in m2 ie it has dimensions oflength square we can write for the area of the whole triangle Ac2fB where fB is some dimensionless function that only depends on the angle B Since similarly we have analogous expressions for the areas of triangles I and II we immediately write czfB azfB bzfB which yields the soughtfor result upon canceling the common factor fB Lecture two August 28 Let us now turn to something else Actually this is an important thing because all modern science started some 400 years ago with this experiment 7 Galileo dropping objects from the leaning tower of Pisa 0 Estimate the height of the tower 56 m 0 Estimate how long it takes a ball to fall a few seconds 0 Use this example to introduce position vector velocity vector speed average velocity instantaneous velocity acceleration 0 Relation between position velocity acceleration o Integrals and derivatives 0 Galileo s hypothesis that balls fall the same independent of mass and general opposition to it Aristotle Philosophy vs Natural Science 0 Derivation and discussion of I h ho 7 gtzZ limiting cases I V 7 g I tfall 2hg12 how does the fall time scale with the height A bit more ballistics Cannon shoots a cannonball at a certain initial speed v0 0 How far does the ball travel in the horizontal direction till it hits the ground 0 At which angle should one tilt the barrel so the cannonball ies the farthest Solve in a couple of different ways introduce optimization using the zero derivative method We can use this problem to once again highlight some general principles of how a mechanics problem may be solved 0 Choose a convenient coordinate frame if convenient write separate eqns for independent motion along different coordinates Are there any limiting cases for which the answer is obvious After the answer is obtained does it have correct dimensions Does it make sense in the limiting cases thought of above 00 Lecture three September 2 The falling stonefeather demonstration in air and in vacuum https0cratesberkeleyedubudkerPhysicsH7A 3 Physics H7A Fall 2003 D Budker The next topic I d like to discuss is rotational motion This is usually discussed somewhat later in the course however I d like to introduce it now in order to illustrate an important concept if you know one thing well you automatically know a whole bunch of other things well We will now see direct analogies between linear and circular motion Other examples are just to give you an idea if you know how a pendulum works really well you also know RLC circuits in electronics waves in the ocean oscillations in plasma the structure of a light beam etc etc 0 Uniform circular motion Linear and angular velocity Angle is the analog of linear coordinate m is analog of v Derivation of vmR re ne to the vector form 7 a7 X R Note that velocity for uniform motion while of constant magnitude is continuously changing direction gt acceleration derive 5 a7 X 7 so that av2R Polar coordinates and vector representation of O and 03 What are the linear velocities of various points on a rolling wheel If a train is moving from Moscow to St Petersburg are there any parts that are moving from St Petersburg to Moscow 000 00 Lecture four September 4 The 31 Newton s Laws 0 Difference in the way physical theory is built cf mathematical axiomatics o The First Law inertial frames 0 The Second Law I m5 what is mass force IT is the vector sum of all forces acting on the particle The Third Law ii b O o The Universal Gravity Law Tim where G 667103911 Nmzkgz remarkably the masses entering this law are the same as in the Second Law the Eguivalence Principle The origins of gravity Gravitational forces due to spherical objects Derivation of the fact that a body of spherical shape exerts gravitational force on an external mass as if all its mass was concentrated in the center Absence of the gravitational force within a spherical shell Observation that problems with simple answers usually have simple solutions Lecture ve September 9 More discussion of gravity forces due to spherically symmetric objects the inside case The simple way to get the result on gravitational forces due to spherical objects is to use the Gauss Theorem The notions of gravitational field the minus sign in the Newton s gravitation law revisited field lines formulation and explanation of the Theorem application to spherical shells https0cratesberkeleyedubudkerPhysicsH7A 4 Physics H7A Fall 2003 D Budker A story about how Richard Feynman became a physicist balls in a Radio Flyer Subtleties of inertial vs noninertial frames We de ned an inertial frame as such a frame where a body does not accelerate in the absence of forces Also in an inertial frame we have the Second Newton s Law Does a frame which is free falling in the Earth s gravitational eld qualify as an inertial frame We can safely say that it does not because the bodies do not accelerate in this frame upon the action of the Earth s gravitation force so the Second Law does not hold However and this is the tricky part as a consequence of the equivalence principal an observer in such a system cannot tell whether they are in an inertial frame or a frame freefalling in the gravitational field unless they see the Earth So from the perspective of such an observer who is ignorant of the fact that there is a body Earth exerting gravitational pool this would seem like a perfectly fine inertial frame Example of the application of the Newton s Laws monkey on a rope Another example rotating conical pendulum stability analysis Demonstration of an inverted pendulum Lecture six September 11 The Foucault pendulum demonstration and discussion Items from last time conical pendulum and stability analysis Demonstration of the conical pendulum 7 how a stableequilibrium point becomes unstable The Coriolis force arising when a body is involved in rotational motion and is changing the radius at the same time 7 a straightforward derivation Lecture seven September 16 Calculation of the Coriolis acceleration for a car moving from SF to LA with v72 kmhr How do we decide whether a quantity is large or small Example the effect of Coriolis forces on weather systems a calculation of forces on air resulting in wind air density Items from previous lectures forces between rope and pulley Momentum generalization of the Second Law Forces acting on composite systems 7 external and internal Momentum conservation some simple examples Demonstrations water in rotating bucket candles on rotating platform Lecture eight September 18 Introduction to rocket Science 7 the Tsialkovskii formula estimate of the launch weight to payload weight ratio for space travel including the estimate of the orbital speed for a low circular orbit around the Earth Does it help much to launch from the equator An example of momentum conservation a fisherman in a boat problem center of mass A more subtle case water friction cx v https0cratesberkeleyedubudkerPhysicsH7A 5 Physics H7A Fall 2003 D Budker Elastic and inelastic collisions demos with balls Experiment with ri e shooting into wooden block veri cation of the momentum conservation law Lecture nine September 23 Demonstration of a compressed air rocket A brief discussion of dry friction How does the car s stopping time depend on its mass Experiments verifying the law of friction FfrmaxlJN where N is the magnitude of the normal force Experiment showing large difference between static and dynamic friction Experiment in which we measure p using the inclined surface method Definition of work and power units derivation of P E 47 Example work and energy conversion when we lift a weight in gravitational eld and then drop it Energy conservation Demonstration of the brachistochrone property of the cycloid Lecture ten September 25 Demonstration of atmospheric pressure 7 collapsing metal can Demonstration of separation of motion in two orthogonal directions 7 shooting a ball vertically from a moving platform Use of energy conservation to calculate velocity in a complicated motion Relation between kinetic energy and work 7 the workenergy theorem More on the brachistochrone An idea on how to minimize path integrals Springs parallel and series connection Potential energy of a spring Demonstrations of the Hooke s law and harmonic oscillation Measuring the dependence or lack thereof of the frequency on the mass and amplitude Derivation of the Simple Harmonic Oscillator SHO motion Energy transformation in a SHO Lecture eleven September 30 Oscillation of two masses connected with a spring reduced mass Other examples of the use of reduced mass planets atoms molecules Oscillations near minimum of a general potential Taylor expansion to obtain SHO approximation Some other examples of SHO pendulum electrical LC circuit Rotational dynamics moment of inertia moments of inertia of some simple configurations Lecture twelve October 2 More examples of moments of inertia The parallel axis theorem Angular momentum and is conservation demonstration https0cratesberkeleyedubudkerPhysicsH7A 6 Physics H7A Fall 2003 D Budker 0 Torque i f as analog of I o Demonstration that solid disc rotates twice as fast as a hollow disc of the same mass and under the same torque Lecture thirteen October 7 o A more detailed discussion of the last demo how can we justify the constant torque approximation the neglect of the moment of inertia of the apparatus etc The problem of a disc on an inclined plane with friction effective inertial mass discussion of energy balance Planetary motion the three Kepler s Laws and how they relate to angular momentum Gyroscopes how can we understand precession from L f a demo Derivation of the gyroscope s precession frequency October 9 Midterm need blue books Lecture fourteen October 14 o The moment of inertia demo discussion continued 7 what went wrong and how can we correct our mistakes How to take square root of 17 The torque demo Why do we say that the force of gravity is applied to the cm The physical pendulum More fun with gyro demos A discussions of the seasons with a demo inclination of the Earth rotation aXis with respect to ecliptic o Precession of the Equinoxes Lecture fteen October 16 o Rotations do not commute A demonstration with a book 0 The Berry s phase 7 an example with a thumb o Feynman s demo with a coffee cup that shows that sometimes you need 2X27 rotations to bring a system to its original state 0 More fun with gyro demos Lecture sixteen October 21 o Nutation derivation of the nutation equations solution limiting cases 0 Fictitious forces in noninertial frames a cylinder on accelerating table 0 Derivation of the cylinder s motion upon action of a force effective mass Lecture seventeen October 23 o Bouncing ball as seen by a stationary observer and an observer riding in an elevator Energy is not invariant with respect to Galilean transformation neither is https0cratesberkeleyedubudkerPhysicsH7A 7 Physics H7A Fall 2003 D Budker whether there is energy exchange between the ball and the wall or not Total energy is however conserved in either frame Fictitious forces in uniformly rotating frames The moon in the swimming pool problem Water surface is an equipotential This discussion turned into a nice sociological experiment that has shown that scientific truth cannot be decided by a democratic vote Three solutions to the problem were offered the correct solution got the smallest number of votes Derivation using the equipotential method and demonstration of the parabolic surface shape of water in a rotating bucket Lecture eighteen October 28 Tides a qualitative discussion and derivation Comparison of the effects of the Sun and the Moon Debunking theories of the Great Flood Fluids Note there is very little on uids in the KampK book We will start with the simplest case of incompressible and inviscid uid Pressure of liquid at a given depth Hydraulic lift Lecture nineteen October 30 Archimedes Law buoyancy Potential energy of uid under pressure 7 the entire static liquid is equipotential Liquid ow in pipes The continuity equation Energy conservation 6 the Bernoulli equation The speed with which water ows from a bucket if a whole is punched in the side Lecture twenty November 4 Viscosity viscous drags balls bubbles etc A discussion of attached mass The Poiseuille ow Turbulence the Reynolds number Rotational ow and vortex motion Lecture twenty one November 6 Oscillations of water in a cup deepwater gravity waves Kelvin s shipwake wedge shallowwater gravity waves General properties of waves 7 m k dispersion relation mk Phase velocity vmk wave packets and group velocity vgdmdk Capillary forces surface tension pressure under curved surface Ap26R For water at room temperature 6 73 mNm Lecture twenty two November 13 Capillary waves dispersion relation group and phase velocities estimate of the cutoff wavelength below which capillary effects are more important for waves than gravity and above which the opposite is true End of Fluids https0cratesberkeleyedubudkerPhysicsH7A 8 Physics H7A Fall 2003 D Budker Lecture twenty three November 18 Central force motion general properties Centrifugal barrier effective potential General equations of motion trajectory Planetary motion Physlet computer simulations 0 O O 0 Lecture twenty four November 20 o More on planetary motion End of central force motion 0 Damped oscillator physical meaning and general solution Lecture twenty ve November 25 o More on damped oscillator The Q factor 0 Forced oscillations Lecture twenty six December 2 0 Acoustic waves 0 Musical instruments Lecture twenty seven December 4 o A special guest lecture Prof Erwin L Hahn on the physics of string musical instruments with lots of demonstrations httpsocratesberkeleyedubudkerPhysicsH7A 9


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