# AP Physics C Mechanics Objectives

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1 AP Physics C Mechanics Objectives I. KINEMATICS A. Motion in One Dimension 1. The relationships among position, velocity and acceleration a. Given a graph of position vs. time, identify or sketch a graph of velocity vs. time b. Given a graph of velocity vs. time, identify or sketch graphs of position and acceleration vs. time c. Given x(t), determine v(t) and a(t), and determine when these quantities have maxima and minima 2. The special case of motion with constant acceleration a. Write down expressions for v(t) and x(t) and identify or sketch graphs of these quantities b. Use the kinematic equations to solve problems in one-dimensional motion with constant acceleration, including ones which involve two accelerating bodies 3. Situations where acceleration is a specific function of velocity and/or time: write down an appropriate differential equation dv / dt = f (v)g(t) and solve it for v(t), incorporating a given initial value of v(t). B. Motion in a Plane 1. Displacement and velocity vectors a. Relate v, x, and t for motion with constant velocity b. Resolve a vector into components along perpendicular axes c. Add vectors d. Subtract vectors to find the location of one particle relative to another 2. The general motion of a particle in two dimensions: Given functions x(t) and y(t) which describe this motion, determine the components, magnitude, and direction of v and a as functions of time 3. Projectile Motion a. Write down expressions for the horizontal and vertical components of v and x as functions of time b. Use these expressions in analyzing the motion of a projectile which is projected with a given initial velocity 4. Uniform Circular Motion a. Relate the radius of the circle, the speed of the particle, and the rate of revolution b. Determine the magnitude and direction of a particle's velocity and acceleration II. NEWTON'S LAWS OF MOTION A. Dynamics of a Single Body 1. The relation between a force which acts on a body and the resulting change in the body's velocity a. For a body moving in one dimension, calculate the change in velocity when a force F(t) acts over a specified time interval b. For a body moving in a plane whose velocity vector undergoes a specified change over a specified time interval, determine the average force F which acted on the body 2. Apply F = ma to forces such as gravity, strings, or contact forces a. Draw well-labeled free-body diagrams showing all real forces acting on a body b. Write down the equation F = ma for components along appropriate axes 3. Static equilibrium under the action of several forces 4. When an object moves with specified acceleration in the earth's gravitational field, analyze situations such as: a. motion up or down with constant acceleration (eg. an elevator) b. motion in a horizontal circle c. motion in a vertical circle 5. Coefficient of friction; understand situations such as a. The relation between normal force and force of friction b. Body sliding down a rough plane or across a rough surface c. Static situations involving friction

2 B. Systems of Two or More Bodies 1. Newton's third law of action and reaction 2. Forces between two bodies which accelerate together or slide across one another 3. Two bodies connected by a string across a frictionless pulley 4. Problems that require solving simultaneous linear equations for forces and accelerations III. WORK, ENERGY AND POWER A. Work and the Work-Energy Theorem 1. The definition of work a. Integrate to find the work done by a variable force F(x) in one dimension b. Use the dot product to find the work done by a constant force in a plane c. Describe, graph or calculate the work done on a body by a non-constant force 2. The work-energy theorem a. State the theorem and prove it for motion in one dimension b. Apply the theorem to find the change in a body's kinetic energy, or the speed which results from the application of a given force B. Conservative Forces and Potential Energy 1. The definition of conservative force 2. The concept of potential energy a. The general relationship between force and potential energy b. Calculate the function U(x) from F(x) or vice versa c. Hooke's law (F = kx) and the energy stored in a compressed spring C. Conservation of Energy 1. Total Mechanical Energy a. The relationship between the work done by a nonconservative force and the change in mechanical energy, and the other forms this energy takes (eg. heat energy) 2. The Conservation of Energy a. Identify situations in which energy is or is not conserved b. Apply conservation of energy to situations where a body moves in a gravitational field and is subject to forces by surfaces, springs or strings 3. Conservation of Energy and Newton's Laws D. Power 1. The definition of power a. Work done by a force which supplies constant power, or average power supplied by a force performing a specified amount of work b. Prove that Power = Force Velocity from the definition of work, and apply the relation to particle motion IV. SYSTEMS OF PARTICLES - STATICS A. Center of Mass 1. Finding the center of mass a. Find by inspection the center of mass of a symmetric body b. Find the center of mass of a system of n particles c. Integrate to find the center of mass of a simple plane or solid body 2. State, prove, and apply the relation between center-of-mass acceleration and net external force

3 3. Express the gravitational potential energy as a function of the position of the center of mass B. Linear Momentum and Collisions 1. The relation between linear momentum and force a. Conservation of linear momentum follows from Newton's third law b. State and apply the relations between linear momentum and center-of-mass motion for a system of particles c. Definition of impulse, and its relation to momentum d. Calculate the force required to hold a fixed body which is emitting, absorbing, or reflecting particles at a certain rate 2. Conservation of linear momentum a. Analyze situations when two bodies are pulled together or pushed apart eg. by a spring, and calculate the energy involved b. In collisions of particles in one or two dimensions, with unknown masses or velocities, calculate the outcome and/ or the energy lost 3. Frames of reference a. Analyze motion relative to a moving medium such as a flowing stream or the wind b. Transform the description of a collision into the center-of-mass frame of reference V. ROTATIONAL MOTION A. Rotational Kinematics 1. By analogy with linear kinematics, write down and apply the equations for angular displacement, velocity and acceleration 2. Know the vector form of the angular quantities and be able to use the right-hand rule B. Rotational Inertia (Moment of Inertia) 1. Recognize by inspection which of a set of symmetric objects has the greatest or least rotational inertia 2. Find the rotational inertia of: a. a collection of point masses in a plane, about an axis perpendicular to the plane b. a thin rod of uniform density, about any axis perpendicular to the rod c. a thin cylindrical shell, or other object which some symmetry C. Rotational Dynamics 1. Fixed-axis a. By analogy with linear dynamics, write down and apply the equations for angular acceleration under the action of an external torque b. Apply conservation of energy to the rotational case 2. Motion of a rigid body along a surface a. Relation between linear and angular velocity or acceleration for a body which rolls without slipping b. Apply the equations of translational and rotational motion simultaneously in analyzing rolling without slipping c. Apply energy conservation to objects rolling without slipping d. Apply rotational dynamics to find and use the torque due to friction when an object rolls, with or without slipping D. Angular Momentum 1. Using vector concepts, calculate: a. torque for a specified force about an arbitrary origin b. angular momentum for a particle moving in a straight line, about an arbitrary origin c. angular momentum for a rotating rigid body in the case where angular momentum is parallel to angular velocity 2. Conservation of angular momentum a. State and use the relation between net external torque and angular momentum, and identify situations where angular momentum is conserved b. Analyze problems where the moment of inertia changes c. Analyze collisions between a moving particle and a rotating rigid body

4 E. Torque and Rotational Equilibrium VI. GRAVITATION 1. State and use the conditions for translational and rotational equilibrium of a rigid body A. Newton's Law of Universal Gravitation 1. Determine the force which one spherically symmetric object exerts on another, and determine the gravitational field at a specified point either inside or outside such an object 2. Derive and use the expression for the (negative) potential energy of a system of two or more bodies interacting gravitationally B. Circular Orbits 1. Find the speed and/or period of an object in a circular orbit by equating the centripetal force to the gravitational force 2. State, prove, and apply Kepler's third law for planetary systems 3. Derive and apply the relations among kinetic energy, potential energy, and total energy for circular orbits C. Elliptical Orbits 1. State Kepler's three laws and describe elliptical orbits in qualitative terms 2. Apply angular momentum conservation to prove Kepler's Second Law, and to relate the speeds of a body at the two extremes of its elliptical orbit 3. Apply energy conservation in analyzing the motion of an object projected straight up from a planet's surface, or which is released from rest far above the surface of the planet VII. OSCILLATIONS A. Kinematics of Simple Harmonic Motion 1. Sketch or identify a graph of displacement vs. time in simple harmonic motion, and identify the amplitude, frequency, and period of the motion 2. Identify points where velocity is a maximum and a minimum, and find velocity as a function of time or displacement 3. Identify points where acceleration is a maximum and a minimum, and find acceleration as a function of time or displacement B. Dynamics of Simple Harmonic Motion 1. Write down the differential equation describing simple harmonic motion, and write down its solution 2. Derive and apply the equation for the period of a mass oscillating on a spring, either vertically or horizontally 3. Determine the effective stiffness constant for springs in "series" or "parallel" combinations, or for pieces of a given spring 4. Derive and apply the equations for the period of a simple pendulum or a torsional (or physical) pendulum C. Energy in Simple Harmonic Motion 1. State how total energy depends on amplitude, and identify points where total energy is either all kinetic or all potential 2. Calculate the kinetic and/or potential energy as a function of time or displacement, and show that the sum of K and U is constant

5 AP Physics C Electricity and Magnetism Objectives I. ELECTROSTATICS A. Charge, Electric Field, and Electric Potential 1. The concept of electric field a. Define electric field in terms of force on a test charge b. Calculate the magnitude and direction of the force on a positive or negative charge placed in a specified field c. Given a diagram of electric lines of force, determine the direction of the field at a given point, identify locations where the field is strong and weak, and identify points where positive or negative charges must be 2. The concept of electric potential a. Calculate the work done on a positive or negative charge which moves through a specified potential difference b. Given a sketch of equipotentials for a given charge distribution, determine the direction and approximate magnitude of the electric field at various positions c. Apply conservation of energy to determine the speed of a charged particle which has been accelerated through a specified potential difference d. Calculate the potential difference between two points in a uniform electric field, and state which is at the higher potential e. Given the electric field strength as a function of position along a line, use integration to find the electric potential as a function of position f. State the general relation between field and potential B. Field and Potential of Point Charges 1. Coulomb's Law and Superposition a. Determine the force which acts between two point charges, and describe the electric field of a single point charge b. Use vector addition to determine the electric field produced by two or more point charges 2. Potential of a point charge a. Determine the electric potential in the vicinity of one or more point charges, and calculate the work needed to move a test charge from one location to another in the vicinity of fixed charges b. Calculate the electrostatic potential energy of a discrete distribution of point charges, and the work needed to change the distribution of charges C. Fields of Charge Distributions 1. The principle of superposition used in integration a. Calculate the field of a straight uniformly charged wire b. Calculate the field or potential of a thin ring of charge on the axis of the ring c. Calculate the potential of a uniformly charged disk on the axis of the disk 2. Highly symmetric fields a. Describe the electric field of an infinite uniformly charged plane, a uniformly charged slab, a long uniformly charged wire, a thin cylindrical shell, or a thin spherical shell b. Use superposition to determine the fields of parallel charged planes, coaxial cylinders, or concentric spheres c. Derive expressions for electric potential as a function of position in the above cases 3. Field and flux a. Calculate the flux of a uniform electric field through an arbitrary surface b. Calculate the flux of an electric field through a curved surface when E is uniform in magnitude and perpendicular to the surface c. Calculate the electric flux through a rectangle when E is perpendicular to the rectangle and is a function of one position coordinate d. Understand the difference between an open surface and a closed surface 4. Gauss's Law a. State the integral form of Gauss's Law b. Use Gauss's Law and symmetry arguments to determine the electric field near a uniformly charged infinite plate, inside or outside a uniformly charged long cylinder or cylindrical shell, and inside or outside a uniformly charged sphere or spherical shell

6 II. ELECTRIC CURRENT AND CIRCUITS A. Current and Resistance 1. The definition of current: Rate of flow of positive charge 2. Resistivity, conductivity and resistance a. Understand the relation between electric field and current density, and the concept of drift velocity of electrons b. Determine the resistance of a resistor of uniform cross-section in terms of its dimensions and the resistivity or conductivity of the material of which it is made, and compare the resistance of resistors of different materials or different geometries c. Derive the expression for the rate at which heat is produced when current passes through a resistor B. Direct-Current Circuits 1. Series and Parallel Combinations of Resistors a. Identify on a circuit diagram which resistors are in series and which are in parallel b. Calculate the equivalent resistance of two or more resistances connected in series or in parallel, or a network of resistors which can be broken down into series and parallel combinations c. Determine the ratio of voltages across resistances in series and parallel or the ratio of currents through resistances in series and parallel d. Calculate the voltage, current, and power dissipation for any resistor in such a network of resistors connected to a single battery 2. EMF and internal resistance of batteries a. Calculate the terminal voltage of a battery of specified EMF and internal resistance through which a known current is flowing b. Calculate the rate at which a battery is supplying energy to a circuit or is being charged up by the circuit c. Determine what external resistance draws maximum power from a battery of specified internal resistance 3. Non-simple circuits a. Apply the loop theorem and the junction theorem to non-simple circuits to generate simultaneous equations for the unknown currents 4. Voltmeters and ammeters a. Understand which is a high-resistance instrument and which is a low-resistance instrument and why b. Assess the effect of connecting non-ideal ammeters or voltmeters on circuits into which they are connected c. Given a meter movement of specified internal resistance and full-scale deflection current, calculate the appropriate shunt or multiplier resistance to create an ammeter or a voltmeter of specified range III. CAPACITANCE AND CAPACITORS A. Electrostatics of Conductors 1. Electric fields in and around conductors a. Explain why there is no electric field inside a conductor carrying no current b. Explain why a conductor is an equipotential, and how this applies to two conductors connected by a wire c. Prove that all excess charge on a conductor must be on its surface and that the electric field just outside a conductor must be perpendicular to its surface d. Use Gauss's Law to relate the surface charge density on a conducting surface to the electric field strength near the surface 2. Induced charge a. Describe qualitatively the process of charging by induction b. Describe the distribution of induced charge on a conducting slab located between two charged parallel plates or on a conducting spherical shell or shells surrounding a point charge, and sketch electric field lines for such cases c. Explain qualitatively why there can be no electric field in a charge-free region completely surrounded by a single conductor, and recognize consequences of this d. Explain qualitatively why the electric field outside a closed conducting surface cannot depend on the precise location of charge in the space enclosed by the conductor, and recognize consequences of this

7 B. Capacitors 1. Definition of capacitance: ratio of charge to voltage 2. Parallel-plate capacitor a. Describe the electric field inside a parallel-plate capacitor, and relate the strength of the field to the potential difference between the plates and the distance between them b. Derive the expression for the capacitance of a parallel-plate capacitor c. Describe how the insertion of a dielectric slab between the plates of a charged parallel-plate capacitor affects its capacitance and the field strength and voltage between the plates d. Analyze situations in which a conducting slab is inserted between the plates of a capacitor 3. Cylindrical and spherical capacitors a. Describe the electric field inside a cylindrical and a spherical capacitor b. Derive the expression for the capacitance of each 4. Work, energy and force in parallel-plate capacitors a. Derive expressions for the energy stored in a capacitor and for the energy density between the plates b. Analyze situations in which parallel capacitor plates are moved apart or moved closer together, or in which a conducting slab is inserted between the plates, either with a battery connected between the plates (holding the voltage constant) or with the charge on the plates held fixed C. Capacitors in circuits 1. Capacitors connected in series or parallel a. Calculate the equivalent capacitance of a series or parallel combination b. Describe how stored charge is divided between two capacitors connected in series or parallel c. Describe the relationship between the voltages across two capacitors in series or parallel 2. Circuits containing capacitors and resistors a. Calculate the voltage and charge for a capacitor connected to a battery and resistors carrying a steady current b. Calculate the time constant of an RC circuit c. Sketch or identify graphs of charge vs. time or voltage vs. time for a capacitor in an RC circuit, and indicate the significance of the time constant d. Write down expressions for the charge or the current as a function of time e. Calculate voltages and currents immediately after a switch is closed in an RC circuit, and also after a long time IV. MAGNETISM A. Forces on Charges and Currents 1. Forces experienced by a charged particle moving in a magnetic field a. Calculate the magnitude and direction of the force in terms of q, v and B and explain why the magnetic force does no work b. Deduce the direction of a magnetic field from information about the forces experienced by charged particles moving through it c. Calculate the radius of the path of a charged particle moving perpendicular to a magnetic field from Newton's Law and the magnetic force law d. Describe the trajectory of a charged particle moving in a magnetic field in the general case e. Describe quantitatively the conditions under which a particle will move with a constant velocity through crossed electric and magnetic fields 2. Forces experienced by a current in a magnetic field a. Calculate the magnitude and direction of the force on a straight segment of current-carrying wire in a uniform magnetic field b. Indicate the direction of magnetic forces on a current-carrying loop of wire in a magnetic field c. Calculate the magnitude and direction of the torque experienced by a rectangular loop of wire carrying a current in a magnetic field B. Production of Magnetic Fields 1. Magnetic field produced by a long straight current-carrying wire a. Calculate the magnitude and direction of the field at any point in the vicinity of the wire b. Calculate the force of attraction or repulsion between two long current-carrying wires

8 2. Biot-Savart Law a. Calculate the magnitude and direction of the contribution to the magnetic field made by a short straight segment of current-carrying wire b. Derive and apply the expression for the magnitude of B on the axis of a circular loop of current 3. Ampere's Law a. Use Ampere's Law and symmetry arguments to derive an expression for the magnitude of B inside or outside a solid or hollow long cylinder carrying a current of uniform density b. Use Ampere's Law to derive an expression for the magnitude of B inside a very long solenoid or inside a toroidal solenoid 4. Apply the superposition principle to find the magnetic field produced by combinations of the above configurations C. Electromagnetic Induction 1. Magnetic Flux a. Calculate the flux of a uniform magnetic field through a loop of arbitrary orientation b. Calculate by integration the flux of a non-uniform magnetic field whose magnitude is a function of one coordinate, through a rectangular loop perpendicular to the field (eg. a loop next to a wire in the same plane) 2. Faraday's Law and Lenz's Law a. Calculate the magnitude and direction of the induced EMF and current in a loop of wire that is being pulled into or out of a uniform magnetic field b. Calculate the magnitude and direction of the induced EMF and current in a loop of wire placed in a spatially uniform magnetic field whose magnitude varies in a specific way with time c. Calculate the magnitude and direction of the induced EMF and current in a loop of wire that rotates at a constant rate about an axis perpendicular to a uniform magnetic field 3. Forces on induced currents a. Analyze situations involving the mechanical consequences of the forces on induced currents 4. Inductance a. Calculate the magnitude and direction of induced EMF in an inductor through which current changes at a specified rate b. Derive and apply the expression for the inductance L of a long solenoid, a pair of current-carrying wires, or a pair of coaxial cylindrical conductors c. Apply Kirchoff s laws to simple LR circuits to obtain a differential equation for the current as a function of time, and solve the differential equation for the current as a function of time d. Determine the initial and steady-state currents in simple or nonsimple LR circuits, and sketch graphs showing current or voltage as a function of time e. Calculate the rate of change of current in an inductor as a function of time, and the energy stored in an inductor with a given current flowing through it f. Understand LC circuits by analogy with a system of mass and spring

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