1. Units. Convert between different units for the same quantity; multiply and divide units of different quantities; and multiply and divide units of the same quantity. Provide proper units with answers.
2. Story. Describe 1-D motion by equations, graphs, and words. Given one type of description, generate any other to describe the same motion. Graphs refer specifically to position-time, velocity-time, and acceleration-time graphs. Graphs may be qualitative or quantitative, depending on the specifications of the problem. Students must be able to generate exact equations for constant-velocity and constant-acceleration motion.
3. Average. Define, distinguish, and apply the quantities of average velocity, instantaneous velocity, average acceleration, and instantaneous acceleration.
4. 1D Kinematics. Relate absolute and relative position, velocity, and time in a 1-D constant-velocity or constant-acceleration situation. This includes creating the position or velocity equation given the other and appropriate initial conditions or other sufficient information; finding the differences between positions and velocities of different constant-velocity objects; and finding the time at which particular events occur.
5. Vectors. Distinguish vectors and scalars. Express vectors graphically, as magnitude and direction, as components, and as linear combinations of unit basis vectors; and convert between these representations. Add, subtract; and take scalar multiples of vectors. Convert between polar and Cartesian coordinate descriptions of vectors. Add and subtract vectors both graphically and by components.
6. Ballistic. Predict, calculate, describe, and relate the horizontal and vertical components of position, velocity, and acceleration to time for a ballistic trajectory without drag. This includes all free-fall projectile problems, including decomposing initial velocity vectors into components, deriving and using the range equation, finding the time at which particular events occur, determining the time and position of the apex of a trajectory, and so on.
7. Circular. Relate period, angular velocity, speed, position, and acceleration for an object undergoing uniform circular motion. This includes specifying the directions of the vectors. It does not include circular motion with changing speed.
8. N1. Equate mechanical equilibrium and zero net force (Newton’s first law).
9. Fbd. Draw a qualitatively correct (appropriate forces with approximate directions and magnitudes) free-body diagram for a body. Applies to objects with balanced or unbalanced forces.
10. Statics. Determine the unknown force in a system in mechanical equilibrium not considering torques. Desired results may be directions or magnitudes or both.
11. Net force. Relate the individual vector forces acting on an object to the net force acting on it. Includes finding the net force from the individual forces, or one component force when the net force is known.
12. N2. Relate an object's net force, acceleration and mass. (Newton's second law.) Includes magnitude and direction of the net force and acceleration.
13. Forces. Determine the magnitudes and directions of the following forces: static and kinetic friction, normal force, tension, uniform-field gravity, viscous drag, and a Hooke’s law spring.
14. N3. Given a force, Identify the object applying it, the object it acts upon, and, its Newton’s third law counterpart.
15. Dot. Calculate dot products of vectors. Identify, recognize, and describe the properties of dot products. The vectors may be given to the student in any format: magnitude and direction, Cartesian components, or linear combinations of unit basis vectors. The student should give the answer in the requested format, which defaults to the format provided.
16. Cross. Calculate cross products of vectors. Identify, recognize, and describe the properties of cross products. Distinguish between dot and cross products. Students are not required to calculate cross products From Cartesian components (the determinant method) unless the angle between the vectors is obvious from the context. Students do need to know the arrangement of the right-handed coordinate system.
17. Work. Relate the work done on an object to the forces applied to it and its displacement during that time. This includes defining and calculating work as the dot product of force and displacement, and recognizing that work is a scalar.
18. K. Calculate the translational kinetic energy of an object from its speed and mass.
19. Work-energy. Relate the net work done on an object to its change in kinetic energy. This applies the work-energy theorem.
20. Potential energy. Calculate an object’s gravitational potential energy (uniform gravitational field) and a Hooke’s law spring’s elastic potential energy. Includes relating the potential energy change to work done against the force or by the force.
21. Conservative forces. Identify and distinguish conservative and non-conservative forces. Inclues identifying and applying the signs of a conservative force: independent of the path, and the negative gradient of a scalar potential energy function.
22. Energy conservation. Use conservation of energy to analyze multi-step processes. Such processes include collisions, ballistic pendulums, Tarzan swinging on a vine, etc. Applies also to situations involving non-conservative forces.
23. Potential Diagrams. Interpret and apply energy diagrams. This includes knowing kinetic and potential energy at any position, relating force to shape of the potential, qualitatively describing a trajectory given starting position and velocity, and explaining the effects of a change in total energy.
24. Σp. Relate the total momentum of a system of objects to the individual masses and velocities of the objects in the system. Includes defining and calculating the momentum of a single object.
25. J-p. Relate the net force on an object, the force’s duration, and the object’s momentum change. This includes defining impulse and applying the impulse-momentum theorem.
26. p Conservation. Apply conservation of momentum to analyze an isolated collision. This includes predicting final velocities, reconstructing initial velocities, and relating whether a collision is elastic, inelastic, or totally inelastic to the characteristics of the collision. It also includes relating initial and final total momentum. Students can relate initial and final velocities of all bodies in a totally inelastic collision in one, two and three dimensions, and of all bodies in an elastic collision in one dimension.
27. Center of mass. Define and calculate the center of mass of a set of point masses, of a composite object, or of an extended object. Relate center of mass, total mass, and total momentum. Know when the center of mass moves and when it doesn’t.
28. Angular kinematics. Relate the angular velocity, angular position, and angular acceleration of a rotor undergoing a constant angular acceleration.
29. Rotor Energy. Determine the rotational kinetic energy of a rigid body. Includes combining translational and rotational kinetic energy (½mv2 + ½Iω2) to find the total kinetic energy of a rigid body. Requires students to recognize any relationship between center-of-mass velocity v and angular velocity ω.
30. Find I. Calculate the moment of inertia of an object from its distribution of mass. Includes adding together segments of known moments of inertia, looking up a moment of inertia formula from a table and applying it properly, applying the parallel-axis theorem, and integrating to find the moment of inertia of a continuous body.
31. Torque. Relate the torques and forces applied to a body, and the net torque to the individual torques. This includes the definition of torque, with full appreciation of its vector nature. It also includes applying the cross product. When adding torques, this requires that they all be evaluated about the same reference point.
32. Angular N2. Relate the net torque on a rotor to the rotor’s moment of inertia and angular acceleration. This applies angular Newton’s second law τ = Iα.
33. Static torques. Find an unknown force, lever arm, or radius vector in a system at mechanical equilibrium.
34. Angular work. Relate the rotational work done on a rotor to applied torque and angular displacement, and to the rotor’s change in rotational kinetic energy. Recognize and apply the formula for angular work W = τΔθ, and apply the work-energy theorem ΔK = W.
35. Angular momentum. Relate a particle’s angular momentum to its momentum and its position relative to a reference point, and a rotor’s angular momentum to its moment of inertia and rotational velocity. Also determine the angular momentum of a body undergoing both translational and rotational motion. These refer to the formulas L = r×p and L = Iω.
36. L conservation. Use conservation of angular momentum to analyze collisions involving rotors, and to predict the motion of an object whose moment of inertia changes. This includes collisions involving objects like swinging doors and pendulums, and to systems such as spinning ice skaters and bolas.
37. SHM kinematics. Relate the position, velocity, acceleration, frequency and period, amplitude, kinetic and potential energies, and phase of an object undergoing simple harmonic motion. Includes symbolically and quantitatively obtaining one equation of motion from another, and determining extreme values of any of the quantities. Includes integrating or differentiating the equations of motion and applying the formulas ω2 = k/m and ΣE = ½ kA2 = ½ mvmax2.
38. SHM energy. Describe the distribution of energy at different phases of the cycle of a simple harmonic oscillator or simple pendulum. Includes recognizing that mechanical energy is conserved, and that amplitude and maximum speed are monotonically related.
39. Simple pendulum. Identify and explain the factors determining the frequency and amplitude of a simple pendulum. Includes applying the formula ω2 = g/L, and recognizing why the displacement angle must be small for the formula to be strictly true.
40. Physical Pendulum. Identify and explain the factors determining the frequency and amplitude of simple and physical pendulums. Calculate the oscillation period of any physical pendulum. Includes recognizing a simple pendulum as a limiting case of a physical pendulum.
41. Waves. Trace, describe, and explain the movement of the medium in mechanical waves. Includes transverse string waves and sound waves.
42. Wave formulas. Relate the mathematical equation of a wave and its period, frequency, amplitude, displacement speed, angular frequency, angular wavenumber, wavelength, and propagation speed. All the parts of y = A cos(kx − ωt + φ).
43. String wave speed. Apply and conceptually explain the relation between length density μ, tension, and propagation speed in a transverse string wave. v2 = F/μ.
44. Decibels. Relate the intensity (watts per square meter) of a sound wave at different distances from the source, and express in terms of decibels. Decibel scale. Convert between SI intensities and decibels.
45. Doppler. Calculate the detected frequency of sound given the source frequency, velocities of source and detector, and speed of sound. Classical Doppler effect; does not include relativistic Doppler shift.
46. Density. Define density; relate density, mass, and volume. Apply the intrinsic nature of density.
47. Pressure. Define pressure; explain how fluids exert pressure on surfaces they contact; relate pressure to force and area.
48. Pascal. Relate pressure changes at different positions in a fluid to each other. Relate pressures, forces, areas, volume changes, and displacements in hydraulic systems. Pascal’s principle and applications, including F1/A1 = F2/A2.
49. Depth. Relate presure in an incompressible fluid to density and depth. p = p0 + ρgh.
50. Buoyancy. Determine the buoyancy force acting on a body partly or completely immersed in a fluid.
51. Bernoulli. Relate speed, pressure, and depth at different locations in an inviscid fluid. Bernoulli's equation.
52. Big G. Calculate the gravitational force between two point masses. Apply Newton’s gravitational formula F = Gm1m2/r2. Specify direction as well as magnitude.
53. Gravitational energy. Calculate the gravitational potential energy between two particles or spheres.
54. Circular orbits. Characterize the radius, speed, period, kinetic energy, potential energy, and total mechanical energy of gravitationally-bound objects in circular orbits. Both in the limit of one object much more massive than the other and in the general case of comparable masses. Includes consequences such as escape speed.
55. Orbit conservation. Use conservation of momentum, mechanical energy, and angular momentum to describe the motion of bodies in all types (circular, elliptical, parabolic, and hyperbolic) of orbit. Includes justifying and applying Kepler’s laws of orbital motion.
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