PHY183B, Summer '99

A. CONTENTS

1. Review of Basic~Mathematical~Skills

2. Physical Description and Measurement

3. Physical Changes and Rates

4. Simple Differentiation and~Integration

5. Vectors

6. Sums, Differences and Products~of~Vectors

7. Kinematics in One Dimension

8. Two-Dimensional Motion

9. Circular Motion: Kinematics

10. An Overview of Particle Dynamics

11. Particle Dynamics; the Laws of Motion

12. Momentum: Conservation and Transfer

13. One-Body Diagrams and Contact Forces

14. Friction in Applications of Newton's Second Law

15. Acceleration and Force in Circular Motion

16. Work, Power, Kinetic Energy

17. Potential Energy, Conservation of Energy

18. Energy Graphs, Motion, Turning Points

19. Dynamics for Circular Motion

20. Torque and Angular Acceleration for Rigid Planar Objects: Flywheels

21. Torque and Angular Momentum in Circular Motion

22. Simple Harmonic Motion

23. Simple Harmonic Motion: Shifted Origin and Phase

24. Newton's Law of Universal Gravitation

25. Orbital Motion in an Inverse-Square-Law Force Field

26. The Wave Equation and Its Solutions

27. Sound Waves and Small Transverse Waves On a String

28. Intensity and Energy in Sound Waves

29. The Doppler Effect

30. Standing Waves in One Dimension

B. THE SKILLS

Review of Basic~Mathematical~Skills

Knowledge

• Vocabulary: real number, magnitude of a number, sign of a number, base, exponent, power, scientific notation, radian, degree, similar triangles, Pythagorean theorem, Cartesian coordinates, acute angle, sine, cosine, tangent, algebraic symbols, algebraic expressions, algebraic quantities.

Rule Application

• Perform elementary arithmetic.
• Calculate the value of a number raised to a power.
• Express numbers in scientific notation and use them in calculations.
• Solve problems using the definitions of "radian" and "similar triangles."
• Use a calculator to calculate common trigonometric functions and their inverses, and, given two appropriate quantities among the angles and sides of a right-angle triangle, calculate other angles and/or sides.
• Solve algebraic expressions.

Physical Description and Measurement

Knowledge

• Vocabulary: length, time, clock, period of a clock, standard, unit, basic standards, basic units, variable, function, independent variable, dependent variable.
• State the condition of unit consistency.
• Describe the SI system of units. State how many basic units it has and give two examples of how these units are defined.
• Define random errors and systematic errors and give an example of each.
• State the difference between the value and the magnitude of a number.

Problem Solving

• Express the value of a given quantity in terms of any specified set of units.
• Given an equation, determine whether its units can be made consistent.
• Given the units of all quantities but one in an equation, find the units of that one.
• Given an arithmetic expression, calculate its correct value with the appropriate number of significant digits and the units in simplest form.
• Given two signed numbers, state which has the larger value and which has the larger magnitude.

Physical Changes and Rates

Knowledge

• Define "change" and state two of its general properties.
• Define "rate," state two of its general properties and give the rates for x = C, x = C t, x = C t².
• Explain what is meant by "small enough" and how it is used to distinguish a difference from a derivative.
• Define "velocity," for motion in a straight line, in these cases: (a) v is constant; (b) v changes uniformly; (c) v is any function.

Problem Solving

• Calculate the average rate of change of a variable, given its value at two points.
• Given the graph of a function, determine the rate of change of the function at any given point.
• Given a function of simple form, find the value or the expression representing its rate of change.
• Given the rate of change of a function which has a constant or uniformly changing rate, calculate total changes in the function.

Simple Differentiation and~Integration

Rule Application

• Differentiate polynomial, exponential, logarithmic, and trigonometric functions.
• Locate the maxima and minima of any given function.
• Solve indefinite integrals of polynomials, exponential, logarithmic, and trigonometric functions.
• Evaluate definite integrals of polynomials, exponential, logarithmic, and trigonometric functions.
• Determine the area under the curve of a given function using the definite integral.

Vectors

Knowledge

• Vocabulary: position, position vector, reference frame, coordinate system, displacement, vector, angle between two vectors, unit vector.
• Define the multiplication of a vector by a number and illustrate with a drawing.
• Define the addition and subtraction of two vectors and illustrate with drawings.

Problem Solving

• Given a quantity, state whether it is a vector or a number, and determine its magnitude.
• Given two vectors, compare their magnitudes and directions and determine whether they are equal.
• Given a vector in one of these forms, represent it in any of the others: (a) algebraic symbol; (b) arrow symbol; (c) a unit vector multiplied by a number.
• Determine the sum or difference of two vectors.
• Determine the product for a given vector multiplied by a number.
• Given a diagram representing the sum or difference of two vectors, write an algebraic equation expressing this relationship.
• Given a vector equation, solve it for any quantity in the equation.

Sums, Differences and Products~of~Vectors

Knowledge

• In two dimensions, add any number of given vectors graphically.
• Given a vector's components in two dimensions, determine the magnitude of the vector and the angle it makes with each of the two coordinate axes, and conversely.
• In three dimensions, add and subtract given vectors expressed as (a) a triad of components and (b) in terms of unit vectors along the coordinate axes.
• Evaluate the scalar product of two given vectors, both in terms of the vectors' components along a fixed set of axes and in terms of the vectors' magnitudes and the angle between them.
• Evaluate the magnitude of the vector product of two given vectors in terms of: their magnitudes and the angle between them. Determine the direction by either the right-hand rule or the screw rule.
• Determine the vector product of two vectors, using their given components along a set of Cartesian coordinate axes.

Kinematics in One Dimension

Knowledge

• Vocabulary: average velocity, instantaneous velocity, speed, average acceleration, instantaneous acceleration.

Problem Solving

• Given a particle's position function as a table, graph or mathematical function of time, determine its average velocity during a specified time interval and its instantaneous velocity at a specified time. Estimate its acceleration during a specified time interval and its instantaneous acceleration at a specified time.
• Given a particle's acceleration function and its velocity and position at specified times, determine its velocity and position at other times.

Two-Dimensional Motion

Knowledge

• Write the vector equations relating position, velocity and acceleration in component form.
• Explain how the motion of component vectors can be used to describe the motion of an object.

Problem Solving

• Given (either graphically or analytically) one of the functions r(t), v(t), or a(t), plus initial conditions, find the two other functions.
• Given r(t), derive the equation of the trajectory for an object.
• Given a special case of constant velocity or constant acceleration for one component, write the appropriate equations of motion by using pre-derived one dimensional relationships.
• Determine the range, maximum height and equation of the trajectory for an object in ballistic motion.

Circular Motion: Kinematics

Knowledge

• Define angular velocity in terms of angular displacement and explain how to determine the direction of the angular velocity vector.

Rule Application

• Given a particle's position (as measured along the arc of its circular path) as a function of time, formally calculate its position, velocity, and acceleration vectors at any instant.
• A particle in uniform circular motion has these four descriptors: radius, speed, radial acceleration (magnitude), plus either of period or frequency. Given any two of these quantities, determine the other two.

An Overview of Particle Dynamics

Knowledge

• Describe the nature of a conservation law, and its use. Give an example.
• State the three conservation laws of particle dynamics; explain all terms and give examples.
• State Newton's three laws: explain the meaning of all terms, and give examples.
• Define momentum and kinetic energy.

Rule Application

• Given an object whose velocity changes in a well-defined situation, determine where the object's increment of momentum and/or kinetic energy came from or went to.

Particle Dynamics; the Laws of Motion

Knowledge

• State the three laws of motion and give simple examples of their application.
• Tell the "Newton and the Horse Who Wouldn't Move" story and explain the flaw in the horse's reasoning.

Problem Solving

• Solve problems involving single objects and constant forces. Some combination of the forces, mass of the object, and the acceleration will be given; you are to find the remaining terms. There are four possible levels of complexity in these problems:
  • (a) the forces all lie in one plane, hence graphical techniques can be used.
  • (b) the acceleration is not given directly; rather the kinematic relations for constant acceleration (two-step problems) will have to be used.
  • (c) The problem is truly three-dimensional, hence it is necessary to work with unit vectors or with component equations.
  • (d) a combination of (b) and (c).
• Solve "person in elevator" type problems.

Momentum: Conservation and Transfer

Knowledge

• Define the momentum of a particle and of a system of particles. State the SI and English units of momentum and express them in terms of force and time units.
• Using Newton's laws and the definition of momentum, show that the momentum of an isolated system of particles does not change with time.

Problem Solving

• Given an object's mass, its velocity before and after a collision, and data relevant to the collision time, calculate the average value of the resultant force on the object during the collision; and, given a collision time and the average resultant force on an object during the collision, calculate the object's change in momentum.
• Given information about two or more particles' momenta before and after a collision, use momentum conservation to determine quantities (speed, directions, etc.) related to the collision.

One-Body Diagrams and Contact Forces

Knowledge

• Vocabulary: contact force, non-contact force, surface force, resultant force.

Rule Application

• Given a non-rotating object on a linear trajectory, or two such objects connected by a string or rope, possibly over a pulley, and given the information on the acceleration of the object and all forces except a surface force, determine the magnitude and direction of the surface force: \begin{one-digit-list}
• (a) Draw a one-body diagram for each object showing the given forces acting on the object, with each force clearly labeled. Resolve those forces into components with one axis along the direction of acceleration and draw those components on another one-body force diagram.
• (b) Write Newton's second law in component form for the known forces and the resultant force.
• (c) Solve for the unknown surface force and add it the the original one-body force diagram. \end{one-digit-list}

Friction in Applications of Newton's Second Law

Knowledge

• Vocabulary: coefficient of non-sliding friction, coefficient of sliding friction, force of non-sliding friction, force of sliding friction.

Rule Application

• Interpret the starting and stopping of motion of given objects such as walking persons, cars, bicycles, boats, and planes in terms of Newton's three laws and forces of non-sliding and sliding friction.

Problem Solving

• Given sufficient information on one or more masses, possibly connected, possibly on inclines, possibly with zero or non-zero non-sliding and/or sliding coefficients of friction, and possibly with constant applied forces, solve for requested values of forces, masses, coefficients of friction, and kinematic variables.

Acceleration and Force in Circular Motion

Knowledge

• Vocabulary: centripetal force, centripetal acceleration, centrifugal force, banking angle, ideal banking angle.
• Derive the expression for the ideal banking angle for uniform circular motion.

Rule Application

• Produce non-gravitational accelerations as numbers times g (called "g's" or "gees").

Problem Solving

• Draw one-body force diagrams for, and solve, problems involving forces, velocity, period, frequency, radius, and mass for an object in uniform circular motion.
• Apply the expression for the ideal banking angle to problems involving uniform circular motion.

Work, Power, Kinetic Energy

Knowledge

• State the line integral definition of the work done by a force and explain how it reduces to other mathematical formulations for special cases.
• Define the power developed by an agent exerting a force.
• Derive the Work-Kinetic Energy Relation using Newton's second law and the work done by a variable force.
• Define the kinetic energy of a particle.

Problem Solving

• Calculate the work done on an object given either:
  • (a) one or more constant forces, or
  • (b) a force that is a function of position along a prescribed path.
• Use the definition of power to solve problems involving agents exerting constant forces on objects moving with constant velocity.
• Use the Work-Kinetic Energy Relation to solve problems involving the motion of particles.

Potential Energy, Conservation of Energy

Knowledge

• Define the potential energy function associated with a conservative force. Explain what a "conservative" force conserves, using examples of both conservative and non-conservative forces.

Problem Solving

• Given any one dimensional conservative force, determine the potential energy function using standard reference points.
• Solve problems involving both conservative and non-conservative forces using the general form of the law of conservation of energy.

Energy Graphs, Motion, Turning Points

Knowledge

• Vocabulary: potential energy curve, energy diagram, turning point.

Problem Solving

• Given the potential energy of a particle as a function of position (in one dimension or radially with spherical symmetry) determine (for a given position) the force acting on that particle and its acceleration, velocity, and turning points (if any).
• Given the graph of a one-dimensional potential energy function and the total energy of a particle, give a qualitative description of the motion of this particle and locate its turning points and regions of acceleration and deceleration.

Dynamics for Circular Motion

Rule Application

• Given a planar configuration of rigidly connected masses and an axis perpendicular to its plane, calculate its moment of inertia about the axis.

Problem Solving

• Use, and justify the use of conservation of angular momentum to solve problems involving torqueless change from one state of uniform circular motion to another.
• Solve problems involving energy of rotation in circular motion, starting from the equation for kinetic energy in linear motion.
• Solve problems involving power transfer in uniform circular motion, starting from the equation for power transfer in uniform linear motion.

Torque and Angular Acceleration for Rigid Planar Objects: Flywheels

Knowledge

• For circular motion, differentiate s = θ r to derive the relation between linear acceleration, angular acceleration, and radius.
• Starting from Newton's second law, derive the relation between torque, moment of inertia, and angular acceleration for a point mass in circular motion.
• For circular motion with constant angular acceleration, derive the general expressions for angular velocity and angular acceleration as functions of time (involves integrals). Check the answers by differentiation.
• For circular motion with constant angular acceleration, eliminate the time variable between the angular displacement and angular velocity expressions to obtain angular velocity as a function of angular displacement and angular acceleration.

Problem Solving

• Solve planar constant-angular-acceleration problems involving torque, moment of inertia, angular velocity, rotational displacement, and time.

Torque and Angular Momentum in Circular Motion

Knowledge

• Define the torque and angular momentum vectors for (a) a single particle (b) a system of particles.
• Starting from Newton's 2nd law, derive its rotational analog and state when it can be written as a scalar equation.
• Start from the equation for linear kinetic energy and derive the corresponding one for rotational kinetic energy.
• Explain why conservation of angular momentum may not hold in one system but may if the system is expanded.

Problem Solving

• For masses in circular motion at fixed radii, solve problems relating torque, moment of inertia, angular acceleration, rotational kinetic energy, work, and angular momentum.
• Given a system in which angular momentum is changing with time due to a specified applied torque, reconstruct the minimum expanded system in which total angular momentum is conserved. Describe the reaction torque which produces the compensating change in angular momentum.

Simple Harmonic Motion

Knowledge

• Vocabulary: oscillatory motion, simple harmonic oscillator, simple harmonic motion, displacement, initial time, amplitude, phase, scaled phase space, frequency, period.
• Write down a general equation for SHM displacement as a function of time, assuming zero initial phase and maximum initial displacement, and identify the amplitude, angular frequency, phase, and displacement. Derive the corresponding equations for velocity and acceleration.

Problem Solving

• For a specific SHO, use given items in this list to produce others, as requested: displacement, time, frequency, period, phase, velocity, angular frequency, acceleration, force, kinetic energy, potential energy, total energy, and word and graphical descriptions of the motion in real space and scaled phase space.

Simple Harmonic Motion: Shifted Origin and Phase

Knowledge

• Define: equilibrium position, displacement from equilibrium position, initial phase, restoring force, linear restoring force, Hooke's law force.
• State Hooke's law, identifying all quantities, and describe the limitations on its validity.
• Start with Newton's second law and Hooke's law and use them to obtain the equation of motion for a simple harmonic oscillator. Find its solutions and derive the relationship of frequency and period to force constant and mass.
• Define angular frequency, frequency and period in terms of the mass m and force constant k for a simple harmonic oscillator.

Problem Solving

• Solve SHM problems involving an equilibrium position that is a finite distance from the origin, and with a finite initial phase. Display solutions as equations, numbers, and/or graphs as requested.

Newton's Law of Universal Gravitation

Knowledge

• Vocabulary: Newton's law of universal gravitation, universal gravitational constant.
• Use Newton's law of universal gravitation, your correctly remembered value of g, and a solar system data table, to calculate the value of the universal gravitational constant G.

Problem Solving

• Calculate the vector gravitational force on one mass due to a given configuration of other masses.

Orbital Motion in an Inverse-Square-Law Force Field

Knowledge

• Vocabulary: apogee, central force, perigee.
• State Kepler's three kinematical laws of planetary motion.
• State how Newton's law of universal gravitation may be used to justify Kepler's laws.

Problem Solving

• Given a table of solar system data, use Newton's law of universal gravitation and the value of g at the earth's surface to determine the period of revolution, orbital speed, and centripetal acceleration of a given circularly orbiting satellite.

The Wave Equation and Its Solutions

Knowledge

• Vocabulary: amplitude, wavelength, wave number, phase, phase constant, wave function, wave speed, wave equation, harmonic function, sinusoidal wave, traveling wave, boundary conditions.
• State the one-dimensional wave equation and its general solution.

Rule Application

• Given a wave function for a one-dimensional traveling wave, verify that it satisfies the wave equation.

Problem Solving

• Given a sufficient number of parameters associated with a sinusoidal wave, write down the mathematical description of the traveling wave.
• Determine the unknown parameters of a one-dimensional sinusoidal wave, given its displacement as a function of either: (i) position at two different times; or (ii) time at two different positions.
• Determine the unknown parameters of a one-dimensional sinusoidal wave, given the wave function and its first derivative with respect to time at x = 0 and t = 0.

Sound Waves and Small Transverse Waves On a String

Knowledge

• Vocabulary: sound wave, longitudinal wave, transverse wave, compressions, rarefactions, bulk modulus (of elasticity), Young's modulus, stress, strain.
• Starting with Newton's second law, derive the expression relating the net force on an element of mass of a stretched string to the transverse acceleration of that string. Comparing the resultant expression to the one-dimensional wave equation, find the expression for the speed of a transverse wave in a stretched string.
• Determine the speed of the waves (in terms of the properties of the medium), given the differential wave equations describing: (i) transverse waves in a stretched string; (ii) longitudinal compressional waves in a solid or a gas.

Rule Application

• For a given harmonic (sinusoidal) disturbance write down the equation representing the waveform and calculate the wavelength and frequency of the wave for: (i) transverse waves in a stretched string; and (ii) longitudinal compressional waves in a solid or a gas.

Intensity and Energy in Sound Waves

Knowledge

• Vocabulary: decibel, plane wave, spherical wave, wave intensity, wave vector, wave front.
• State the expression for the intensity of a plane wave and the power transmitted across a given cross-sectional area of the medium through which the wave travels.

Rule Application

• For intensity calculations associated with plane and spherical waves, use the unit of intensity level, the decibel.

Problem Solving

• For an acoustic plane wave of given amplitude and frequency, traveling through an elastic medium of given mass density and elasticity, calculate the wave's intensity, energy density, and rate at which it propagates energy.
• Given the intensity of a spherical wave at one radial distance from the wave source, calculate the wave's intensity at any other radial distance.

The Doppler Effect

Knowledge

• Vocabulary: Doppler broadening, Doppler effect, Doppler shift.
• Describe how the Doppler effect is used by astronomers and cosmologists to justify the "expanding universe" model.

Problem Solving

• Solve any Doppler shift problem by deriving the shift for that particular case (not by using the Doppler shift formula and not by deriving the general case and then using it).
• Use the Doppler shift formula to determine the Doppler shift for given motions of a sound wave source and a receiver relative to each other as well as to the acoustic medium.
• Given a value for the Doppler shift, calculate the relative speed between receiver and source.

Standing Waves in One Dimension

Knowledge

• Vocabulary: antinode, characteristic (resonant) frequency, first overtone, fundamental frequency, harmonics, node, normal modes, overtones, standing wave, superposition.

Problem Solving

• Given a set of physical boundary conditions, determine the possible frequencies of the normal modes for transverse standing waves on a stretched string. For a string of given mass per unit length under a given tension, determine the numerical values for some of these frequencies.
• For longitudinal sound waves in a closed organ pipe, apply the appropriate boundary conditions to determine the possible frequencies for standing waves. Do the same for an open pipe. Determine the numerical values of the fundamental frequency and some of its lower harmonics.