Course Objectives for Advanced Placement Physics C – Newtonian Mechanics

A. Kinematics

1. Motion in One Dimension

a. Students should understand the general relationships among position, velocity, and acceleration for the motion of a particle along a straight line, so that:

(1) Given a graph of one of the kinematic quantities, position, velocity, or acceleration, as a function of time, they can recognize in what time intervals the other two are positive, negative, or zero, and can identify or sketch a graph of each as a function of time.

(2) Given an expression for one of the kinematic quantities, position, velocity, or acceleration, as a function of time, they can determine the other two as a function of time, and find when these quantities are zero or achieve their maximum and minimum values.

b. Students should understand the special case of motion with constant acceleration so that they can:

(1) Write down expressions for velocity and position as functions of time, and identify or sketch graphs of these quantities.

(2) Use the equations v = v_o + at, x = x_o + v_ot + at²/2, and v² – v_o² = 2a(x – x_o) to solve problems involving one-dimensional motion with constant acceleration.

c. Students should know how to deal with situations in which acceleration is a specified function of velocity and time so they can write an appropriate differential equation dv/dt = f(v)g(t) and solve it for v(t) incorporating correctly a given initial value.

2. Motion in Two Dimensions

a. Students should know how to deal with displacement and velocity vectors so they can:

(1) Relate velocity, displacement, and time for motion with constant velocity.

(2) Calculate the component of a vector along a specified axis, or resolve a vector into components along two specified mutually perpendicular axes.

(3) Add vectors in order to find the net displacement of a particle that undergoes successive straight-line displacements.

(4) Subtract displacement vectors in order to find the location of one particle relative to another, or calculate the average velocity of a particle.

(5) Add or subtract velocity vectors in order to calculate the velocity change or average acceleration of a particle, or the velocity of one particle relative to another.

b. Students should understand the general motion of a particle in two dimensions so that, given functions x(t) and y(t) which describe this motion, they can determine the components, magnitude, and direction of the particle’s velocity and acceleration as functions of time.

c. Students should understand the motion of projectiles in a uniform gravitational field so they can:

(1) Write down expressions for the horizontal and vertical components of velocity and position as functions of time, and sketch or identify graphs of these components.

(2) Use these expressions in analyzing the motion of a projectile that is projected above level ground with a specified initial velocity.

d. Students should understand the uniform circular motion of a particle so they can:

(1) Relate the radius of the circle and the speed or rate of revolution of the particle to the magnitude of the centripetal acceleration.

(2) Describe the direction of the particle’s velocity and acceleration at any instant during the motion.

(3) Determine the components of the velocity and acceleration vectors at any instant, and sketch or identify graphs of these quantities.

B. Newton’s Laws of Motion

1. Static Equilibrium (First Law)
Students should be able to analyze situations in which a particle remains at rest, or moves with constant velocity, under the influence of several forces.

2. Dynamics of a Single Body (Second Law)

a. Students should understand the relation between the force that acts on a body and the resulting change in the body’s velocity so they can:

(1) Calculate, for a body moving in one direction, the velocity change that results when a constant fore F acts over a specified time interval.

(2) Calculate, for a body moving in one dimension, the velocity change that results when a force F(t) acts over a specified time interval.

(3) Determine, for a body moving in a plane whose velocity vector undergoes a specified change over a specified time interval, the average force that acted on the body.

b. Students should understand how Newton’s Second Law, F = ma, applies to a body subject to forces such as gravity, the pull of strings, or contact forces, so they can:

(1) Draw a well-labeled diagram showing all real forces that act on the body.

(2) Write down the vector equation that results from applying Newton’s Second Law to the body, and take components of this equation along appropriate axes.

c. Students should be able to analyze situations in which a body moves with specified acceleration under the influence of one or more forces so they can determine the magnitude and direction of the net force, or of one of the forces that makes up the net force, in situations such as the following:

(1) Motion up or down with constant acceleration (in an elevator, for example).

(2) Motion in a horizontal circle (e.g. mass on a rotating merry-go-round or car rounding a banked curve).

(3) Motion in a vertical circle (e.g. mass swinging on the end of a string, cart rolling down a curved track, rider on a Ferris wheel).

d. Students should understand the significance of the coefficient of friction so they can:

(1) Write down the relationship between the normal and frictional forces on a surface.

(2) Analyze situations in which a body slides down a rough inclined plane or is pulled or pushed across a rough surface.

(3) Analyze static situations involving friction to determine under what circumstances a body will start to slip, or to calculate the magnitude of the force of static friction.

e. Students should understand the effect of fluid friction on the motion of a body so they can:

(1) Find the terminal velocity of a body moving vertically through a fluid that exerts a retarding force proportional to velocity.

(2) Describe qualitatively, with the aid of graphs, the acceleration, velocity, and displacement of such a particle when it is released from rest or is projected vertically with specified initial velocity.

3. Systems of Two or More Bodies (Third Law)

a. Students should understand Newton’s Third Law so that, for a given force, they can identify the body on which the reaction force acts and state the magnitude and direction of this reaction.

b. Students should be able to apply Newton’s Third Law in analyzing the force of contact between two bodies that accelerate together along a horizontal or vertical line, or between two surfaces that slide across one another.

c. Students should know that the tension is constant in a light string that passes over a massless pulley and should be able to use this fact in analyzing the motion of a system of two bodies joined by a string.

d. Students should be able to solve problems in which application of Newton’s Laws leads to two or three simultaneous linear equations involving unknown forces or accelerations.

C. Work, Energy, and Power

1. Work and the Work-Energy Theorem

a. Students should understand the definition of work so they can:

(1) Calculate the work done by a specified constant force on a body that undergoes a specified displacement.

(2) Relate the work done by a force to the area under a graph of force as a function of position, and calculate this work in the case where the force is a linear function of position.

(3) Use integration to calculate the work performed by a force F(x) on a body that undergoes a specified displacement in one dimension.

(4) Use the scalar product operation to calculate the work performed by a specified constant force F on a body that undergoes a displacement in a plane.

b. Students should understand the work-energy theorem so they can:

(1) State the theorem precisely, and prove it for the case of motion in one dimension.

(2) Calculate the change in kinetic energy or speed that results from performing a specified amount of work on a body.

(3) Calculate the work performed by the net force, or by each of the forces that makes up the net force, on a body that undergoes a specified change in speed or kinetic energy.

(4) Apply the theorem to determine the change in a body’s kinetic energy and speed that results from the application of specified forces, or to determine the force that is required in order to bring a body to rest in a specified distance.

2. Conservative Forces and Potential Energy

a. Students should understand the concept of a conservative force so they can:

(1) State two alternative definitions of “conservative force,” and explain why these definitions are equivalent.

(2) Describe two examples each of conservative forces and nonconservative forces.

b. Students should understand the concept of potential energy so they can:

(1) State the general relation between force and potential energy, and explain why potential energy can be associated only with conservative forces.

(2) Calculate a potential energy function associated with a specified one dimensional force F(x).

(3) Given the potential energy function U(x) for a one-dimensional force, calculate the magnitude and direction of the force.

(4) Write an expression for the force exerted by an ideal spring and for the potential energy stored in a stretched or compressed spring.

(5) Calculate the potential energy of a single body in a uniform gravitational field.

(6) Calculate the potential energy of a system of bodies in a uniform gravitational field.

(7) State the generalized work-energy theorem and use it to relate the work done by nonconservative forces on a body to the changes in kinetic and potential energy of the body.

3. Conservation of Energy

a. Students should understand the concepts of mechanical energy and of total energy so they can:

(1) State, prove, and apply the relation between the work performed on a body by nonconservative forces and the change in a body’s mechanical energy.

(2) Describe and identify situations in which mechanical energy is converted to other forms of energy.

(3) Analyze situations in which a body’s mechanical energy is changed by friction or by a specified externally applied force.

b. Students should understand conservation of energy so they can:

(1) Identify situations in which mechanical energy is or is not conserved.

(2) Apply conservation of energy in analyzing the motion of bodies that are moving in a gravitational field and are subject to constraints imposed by strings or surfaces.

(3) Apply conservation of energy in analyzing the motion of bodies that move under the influence of springs.

(4) Apply conservation of energy in analyzing the motion of bodies that move under the influence of other specified one-dimensional forces.

c. Students should be able to recognize and solve problems that call for application both of conservation of energy and Newton’s Laws.

4. Power

a. Students should understand the definition of power so they can:

(1) Calculate the power required to maintain the motion of a body with constant acceleration (e.g. to move a body along a level surface, to raise a body at a constant rate, or to overcome friction for a body that is moving at a constant speed).

(2) Calculate the work performed by a force that supplies constant power, or the average power supplied by a force that performs a specified amount of work.

(3) Prove that the relation P = F • v follows from the definition of work, and apply this relation in analyzing particle motion.

D. Systems of Particles, Linear Momentum

1. Center of Mass

a. Students should understand the technique for finding center of mass so they can:

(1) Identify by inspection the center of mass of a body that has a point of symmetry.

(2) Locate the center of mass of a system consisting of two such bodies.

(3) Use integration to find the center of mass of a thin rod of nonuniform density, of a plane lamina of uniform density, or of a solid of revolution of uniform density.

b. Students should be able to state, prove, and apply the relation between center-of-mass velocity and linear momentum, and between center-of-mass acceleration and net external force for a system of particles.

c. Students should be able to define center of gravity and to use this concept to express the gravitational potential energy of a rigid body in terms of the position of its center of mass.

2. Impulse and Momentum:  Students should understand impulse and linear momentum so they can:

a. Relate mass, velocity, and linear momentum for a moving body, and calculate the total linear momentum of a system of bodies.

b. Relate impulse to the change in linear momentum and the average force acting on a body.

c. State and apply the relations between linear momentum and center-of-mass motion for a system of particles.

d. Define impulse, and prove and apply the relation between impulse and momentum.

3. Conservation of Linear Momentum, Collisions

a. Students should understand linear momentum conservation so they can:

(1) Explain how linear momentum conservation follows as a consequence of Newton’s Third Law for an isolated system.

(2) Identify situations in which linear momentum, or a component of the linear momentum vector, is conserved.

(3) Apply linear momentum conservation to determine the final velocity when two bodies that are moving along the same line, or at right angles, collide and stick together, and calculate how much kinetic energy is lost in such a situation.

(4) Analyze collisions of particles in one or two dimensions to determine unknown masses or velocities, and calculate how much kinetic energy is lost in a collision.

(5) Analyze situations in which two bodies are pushed apart by a spring or other agency, and calculate how much energy is released in such a process.

b. Students should understand frames of reference so they can:

(1) Analyze the uniform motion of a particle relative to a moving medium such as a flowing stream.

(2) Transform the description of a collision or decay process to or from a frame of reference in which the center of mass of the system is at rest.

(3) Analyze the motion of particles relative to a frame of reference that is accelerating horizontally or vertically at a uniform rate.

E. Rotation

1. Torque and Rotational Statics

a. Students should understand the concept of torque so they can:

(1) Calculate the magnitude and sense of the torque associated with a given force.

(2) Calculate the torque on a rigid body due to gravity.

b. Students should be able to analyze problems in statics so they can:

(1) State the conditions for translational and rotational equilibrium of a rigid body.

(2) Apply these conditions in analyzing the equilibrium of a rigid body under the combined influence of a number of coplanar forces applied at different locations.

2. Rotational Kinematics

a. Students should understand the analogy between translational and rotational kinematics so they can write and apply relations among the angular acceleration, angular velocity, and angular displacement of a body that rotates about a fixed axis with constant angular acceleration.

b. Students should be able to use the right-hand rule to associate an angular velocity vector with a rotating body.

3. Rotational Inertia

a. Students should develop a qualitative understanding of rotational inertia so they can:

(1) Determine by inspection which of a set of symmetric bodies of equal mass has the greatest rotational inertia.

(2) Determine by what factor a body’s rotational inertia changes if all its dimensions are increased by the same factor.

b. Students should develop skill in computing rotational inertia so they can find the rotational inertia of:

(1) A collection of point masses lying in a plane about an axis perpendicular to the plane.

(2) A thin rod of uniform density, about an arbitrary axis perpendicular to the rod.

(3) A thin cylindrical shell about its axis, or a body that may be viewed as being made up of coaxial shells.

(4) A solid sphere of uniform density about an axis through its center.

c. Students should be able to state and apply the parallel-axis theorem.

4. Rotational Dynamics

a. Students should understand the dynamics of fixed-axis rotation so they can:

(1) Describe in detail the analogy between fixed-axis rotation and straight-line translation.

(2) Determine the angular acceleration with which a rigid body is accelerated about a fixed axis when subjected to a specified external torque or force.

(3) Apply conservation of energy to problems of fixed-axis rotation.

(4) Analyze problems involving strings and massive pulleys.

b. Students should understand the motion of a rigid body along a surface so they can:

(1) Write down, justify, and apply the relation between linear and angular velocity, or between linear and angular acceleration, for a body of circular cross-section that rolls without slipping along a fixed plane, and determine the velocity and acceleration of an arbitrary point on such a body.

(2) Apply the equations of translational and rotational motion simultaneously in analyzing rolling with slipping.

(3) Calculate the total kinetic energy of a body that is undergoing both translational and rotational motion, and apply energy conservation in analyzing such motion.

5. Angular Momentum and Its Conservation

a. Students should be able to use the vector product and the right-hand rule so they can:

(1) Calculate the torque of a specified force about an arbitrary origin.

(2) Calculate the angular momentum vector for a moving particle.

(3) Calculate the angular momentum vector for a rotating rigid body in simple cases where this vector lies parallel to the angular velocity vector.

b. Students should understand angular momentum conservation so they can:

(1) Recognize the conditions under which the law of conservation is applicable and relate this law to one- and two-particle systems such as satellite orbits or the Bohr atom.

(2) State the relation between net external torque and angular momentum, and identify situations in which angular momentum is conserved.

(3) Analyze problems in which the moment of inertia of a body is changed as it rotates freely about a fixed axis.

(4) Analyze a collision between a moving particle and a rigid body that can rotate about a fixed axis or about its center of mass.

F. Oscillations

1. Students should understand the kinematics of simple harmonic motion so they can:

a. Sketch or identify a graph of displacement as a function of time, and determine from such a graph the amplitude, period, and frequency of the motion.

b. Write down an appropriate expression for displacement of the form A sin wt  or  A cos wt  to describe the motion.

c. Identify points in the motion where the velocity is zero or achieves its maximum positive or negative value.

d. Find an expression for velocity as a function of time

e. State qualitatively the relation between acceleration and displacement.

f. Identify points in the motion where the acceleration is zero or achieves its greatest positive or negative value.

g. State and prove the relation between acceleration and displacement.

h. State and apply the relation between frequency and period.

i. Recognize that a system that obeys a differential equation of the form d²x/dt² = - kx must execute simple harmonic motion, and determine the frequency and period of such motion.

j. State how the total energy of an oscillating system depends on the amplitude of the motion, sketch or identify a graph of kinetic or potential energy as a function of time, and identify points in the motion where this energy is all potential or all kinetic.

k. Calculate the kinetic and potential energies of an oscillating system as functions of time, sketch or identify graphs of these functions, and prove that the sum of kinetic and potential energy is constant.

l. Calculate the maximum displacement or velocity of a particle that moves in simple harmonic motion with specified initial position and velocity.

m. Develop a qualitative understanding of resonance so they can identify situations in which a system will resonate in response to a sinusoidal external force.

2. Students should be able to apply their knowledge of simple harmonic motion to the case of a mass on a spring, so they can:

a. Derive the expression for the period of oscillation of a mass on a spring.

b. Apply the expression for the period of oscillation of a mass on a spring.

c. Analyze problems in which a mass hangs from a spring and oscillates vertically.

d. Analyze problems in which a mass attached to a spring oscillates horizontally.

e. Determine the period of oscillation for systems involving series or parallel combinations of identical springs, or springs of differing lengths.

3. Students should be able to apply their knowledge of simple harmonic motion to the case of a pendulum, so they can:

a. Derive the expression for the period of a simple pendulum.

b. Apply the expression for the period of a simple pendulum.

c. State what approximation must be made in deriving the period.

d. Analyze the motion of a torsional pendulum or physical pendulum in order to determine the period of small oscillations.

G. Gravitation

1. Students should know Newton’s Law of Universal Gravitation so they can:

a. Determine the force that one spherically symmetrical mass exerts on another.

b. Determine the strength of the gravitational field at a specified point outside a spherically symmetrical mass.

c. Describe the gravitational force inside and outside a uniform sphere, and calculate how the field at the surface depends on the radius and density of the sphere.

2. Students should understand the motion of a body in orbit under the influence of gravitational forces so they can:

a. For a circular orbit:

(1) Recognize that the motion does not depend on the body’s mass, describe qualitatively how the velocity, period of revolution, and centripetal acceleration depend upon the radius of the orbit, and derive expressions for the velocity and period of revolution in such an orbit.

(2) Prove that Kepler’s Third Law must hold for this special case.

(3) Derive and apply the relations among kinetic energy, potential energy, and total energy for such an orbit.

b. For a general orbit:

(1) State Kepler’s three laws of planetary motion and use them to describe in qualitative terms the motion of a body in an elliptic orbit.

(2) Apply conservation of angular momentum to determine the velocity and radial distance at any point in the orbit.

(3) Apply angular momentum conservation and energy conservation to relate the speeds of a body at the two extremes of an elliptic orbit.

(4) Apply energy conservation in analyzing the motion of a body that is projected straight up from a planet’s surface or that is projected directly toward the planet from far above the surface.