Gateways to Content of WIKItextBook

This page offers multiple gateways into the WIKItextBook, each helpful to one or another aspect of learning.  It allows you to view the material in various sharply focused ways: as a hierarchy of models, a glossary, by common interactions, or as a presorted collection of worked examples that all use the MAPS problem solving rubic.  We also offer more traditional chapter-like organizations each called a "learning path" and arranged into a series of lessons.  These reorganizations take advantage of the electronic textBook's intrinsic rearrangeability.

I.  Hierarchy of Models

The central organizing scheme of the WIKI for experienced students. Models bring together the systems, interactions, principles and examples into one succinct package.
    Motion, Acceleration and Net Force - 3D
    Momentum and External Force
    Mechanical Energy, External Work, and Internal Non-Conservative Work
    Angular Momentum and External Torque about a Single Axis
    • Motion, Acceleration and Net Force - 3D — System: One point particle. — Interactions: Any.
      • Two-Dimensional Motion (General) — System: One point particle confined to a plane. — Interactions: Any that respect the planar motion.
        • Rotational Motion — System: One rigid body in pure rotation or one point particle constrained to move in a circle. — Interactions: Any angular acceleration. — Warning: The constraint of rotational motion implies centripetal acceleration may have to be considered.
          • Uniform Circular Motion — System: One point particle constrained to move in a circle at constant speed. — Interactions: Centripetal acceleration.
        • One-Dimensional Motion (General) — System: One point particle constrained to move in one dimension. — Interactions: Any that respect the one-dimensional motion.
          • Simple Harmonic Motion — System: One point particle constrained to move in one dimension. — Interactions: The particle must experience a force (or torque) that attempts to restore it to equilibrium and is directly proportional to its displacement from that equilibrium.
          • 1-D Motion (Constant Acceleration) — System: One point particle moving in one dimension either because it's constrained to move that way or because only one Cartesian component is considered. — Interactions: Constant force (in magnitude or in its component along the axis). 
    • Momentum and External Force — System: Any. — Interactions: Any. — Note: Linear momentum evolves separately from angular momentum, so all system constituents are treated as point particles in this model.
      • Point Particle Dynamics — System: Any system can be treated as a point particle located at the center of mass. — Interactions: Any.
    • Mechanical Energy, External Work, and Internal Non-Conservative Work — System: Any system that does not undergo significant changes in internal energy. — Interactions: Any interactions that can be parameterized as mechanical work. Notable exceptions include heat transfer or radiation.
    • Angular Momentum and External Torque about a Single Axis — System: Any number of rigid bodies or point particles whose angular momentum is constrained to lie along a certain axis. — Interactions: Any that respect the one-dimensional angular momentum.
      • Single-Axis Rotation of a Rigid Body — System: One rigid body rotating about a fixed axis or rotating and translating such that its angular momentum is constrained to one-dimension and the moment of inertia about its center of mass is constant. — Interactions: Any that respect the one-dimensional angular momentum.

    II. Interaction Glossary

    A glossary of the specific interactions that are commonly encountered in mechanics. In addition to being a resource describing these interactions, it is reasonable to consider this glossary to provide an organizational scheme for the WIKI content. From this perspective, the glossary encourages the view that mechanics is a collection of different ways to describe the effects of a particular group of interactions.

      Commonly Encountered Interactions

      Commonly Encountered Interactions

      • contact force — A force that arises when one macroscopic body presses against another.
        • applied force — When a person, animal or machine purposely pushes or pulls an object, the resulting contact force is often called an applied force.
        • collision forces — Contact forces occuring between objects involved in a collision. During a collision between relatively rigid bodies the collision forces will often far exceed other forces on the objects involved in the collision. In the limit that only collision forces are relevant the momentum of the system composed of all the colliding objects is conserved during the collision.
        • friction — The component of the contact force from a surface that is parallel to the plane of the surface. Friction forces will arise when (1.) an object is sliding along the surface or (2.) an object resting on the surface is subject to forces that would cause it to slide in the absence of friction. The force of friction will always resist the existing or intended sliding motion.
          • kinetic friction — The specific manifestation of friction that is directly opposed to an object's sliding motion along a surface. The force of kinetic friction has a size independent of the speed of the object, and proportional to the normal force exerted on the object by the surface.
          • static friction — The specific manifestation of friction which attempts to resist efforts to move an object that is currently at rest with respect to a surface. If possible, static friction provides just enough force to keep the object stationary, and no more. When the net force attempting to create sliding motion exceeds a certain limiting value proportional to the normal force exerted by the surface on the object, static friction will be unable to prevent motion.
        • normal force — An object in contact with a surface will always be subject to a contact force that has a component perpendicular to the surface which guarantees that the object will not pass through the surface. The name is derived from the mathematical meaning of normal (perpendicular).
      • gravitation (universal) — An interaction between two massive particles resulting in an attractive force exerted on each by the other.  The force is proportional to the gravitational constant G = 6.674 28(67) x 10-11 m3 kg-1 s-2, and the masses of the bodies, and inversely proportional to the square of the distance between them.
        • gravity (near-earth) — The gravitational force exerted by the earth on an object near the earth's surface.
      • Hooke's Law for elastic interactions — A mathematical approximation to the restoring behavior of springs and other elastic solids under small deformations.

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      III.  Vocabulary Glossary

      A collection of the specialized vocabulary that you will need to learn with definitions and explanations.  The alphabetical order of the glossary has no physics-based logic, but might be a good order for reviewing the material.

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            • acceleration — The time rate of change of velocity of an object, or alternately the net force on the object divided by the object's mass.
            • amplitude — The magnitude of the maximum displacement from the rest position of an oscillating system.
            • angular acceleration — The rate of change of the angular velocity with time, or the second derivative of the angular position with respect to time. For systems rotating about a single axis with a fixed moment of inertia about that axis, the angular acceleration is directly proportional to the net torque acting on the system.
            • angular frequency — The magnitude of the angular velocity vector, ω. An angular frequency can also be defined for periodic linear motions like Simple Harmonic Motion by multiplying the ordinary frequency f by 2π (ω = 2πf).
            • angular impulse — the angular impulse is the integral of torque (single-axis) over the time it acts.
            • angular momentum about a single axis — The circulation of linear momentum about the specified axis, being proportional to the component of momentum or each mass along a circle about the axis and the radius of the circle. Angular momentum is changed by external torques, and therefore is constant when these sum to zero. The angular momentum of a rigid body is proportional to its moment of inertia times its angular velocity.
            • angular position — The angular coordinate of a location in polar coordinates, generally represented by the small Greek letter theta, θ .
            • angular velocity — The change in angular position with time, the angular analogue of linear velocity. It is a vector, having both magnitude and direction. In introductory mechanics we will almost always deal with cases of angular velocity about a single axis of rotation, so that the angular velocity is confined to one dimension.
            • axis of rotation — An imaginary line chosen by the problem solver that is perpendicular to the plane of motion of a system and about which angular momenta and torques are calculated.
            • center of mass — The average position of the mass in a body or system.  A system will behave in response to external forces applied to any of its parts as if the entire mass of the body were concentrated there.  The motion of the center of mass is unaffected by internal forces in the system (e.g. forces between the atoms, or collisions between different components of the system).
            • centripetal acceleration — The acceleration directed toward the center of rotation that results from the change in direction (not magnitude) of the velocity when an object is in circular motion.
            • coefficient of friction — Actually two different, but related, constants of proportionality, relating frictional force to the associated normal force
            • conservative force — A force which has an associated potential energy. In introductory mechanics, the only conservative forces generally encountered are gravitation (universal) and elastic forces which satisfy Hooke's Law for elastic interactions.
            • conserved — A quantity that is constant in time (does not change) is said to be conserved.
            • coordinate system — A set of mathematical axes which serve as a quantitative map grid, allowing precise specification of positions of objects.  Cartesian coordinates are most common in introductory mechanics, but cylindrical coordinates are sometimes useful, especially for circular or orbital motion.
            • cross product — Also known as the vector product, the cross product is a way of multiplying two vectors to yield another vector.
            • decomposition — Decomposing a problem into parts, each amenable to solution using the S.I.M. approach.
            • Delta-v diagram — A graphical approach to understanding the form of the centripetal acceleration.
            • displacement — change in position of an object from a fixed reference point.
            • distance — distance is a scalar measurement of the change in location of something from a fixed reference point. It differs from the displacement, which is a vector measurement of the change in location.
            • dot product — A common term for the scalar product, since the scalar product is symbolically indicated by placing a dot between the two vectors being multiplied
            • dynamics — The branch of mechanics that is the study of the interplay between the applied forces on a mechanical body and the resulting motion.
            • elastic collision — A collision in which the momentum and kinetic energy of the system consisting of all objects participating in the collision remains constant.
            • elementary fermionic particle — Thought to be the building blocks of all matter, the fermionic particles currently believed to be elementary (indivisible) are quarks and leptons.
            • environment — The things that can interact with a system and influence its behavior, but which are not directly part of the system and are not modeled.
            • equilibrium position — A stable position in which all forces are balanced (vector sum is zero) and the object of interest is not in motion.  Equilibria may be stable or unstable depending on whether the force acts back toward or away from the equilibrium position if the particle is slightly displaced from in.
            • experiment — Careful observations of constructed situations that have the ability to falsify Laws, measure parameters of the laws, or properties of physical objects.
            • external force — A force exerted on a constituent of a system by the environment.
            • fixed axis — A situation in which the axis of rotation is physically constrained to be in a set location relative to the moving parts, usually because it is fabricated as an axle.
            • force — Force produces a change in the momentum of a mass on which it acts, according to F=ma (Newton's Second Law). Forces result from various types of physical interactions, which always generate a pair of opposite forces acting on two different objects (Newton's Third Law).
            • force diagram — a schematic drawing showing the object under consideration (often represented as a point mass) and the forces acting upon it. the forces are represented as vectors and are labeled.
            • free body diagram — A graphical representation used to analyze the forces exerted on a single system by its environment.
            • freefall — An object that is subject only to the force of gravity is in freefall.
            • friction (interaction)
            • fundamental forces — Forces which can influence the motion of at least one class of elementary fermionic particle. The only fundamental force which is studied directly in introductory mechanics is gravity.
            • gee — gee , or g , is a measure of acceleration equal to the acceleration due to gravity (interaction) at the earth's surface.
            • gravity (interaction)
            • gyroscope — A rapidly-spinning symmetrical top usually used to maintain direction or to demonstrate the principles of angular momentum. Often one treats it with the "gyroscopic approximation" which assumes that the angular momentum is parallel to the direction of spin (i.e. the contribution from precession is assumed negligible).
            • impulse — The time integral of force. The net external impulse acting on a system over a given time interval is equal to the system's change in momentum.
            • inertial reference frame — A frame of reference with respect to which an object with no real forces acting on it will move with constant velocity, i.e. no acceleration.  Newton's Second Law applies only in inertial reference frames.
            • infinitely massive object — Ideally, an object which is the result of letting the mass approach the limiting value of infinity. In practice, an object whose mass so far exceeds those of any others in the system that its mass may, for practical purposes, be taken as that limiting case.
            • initial-state final-state diagram — A diagram illustrating the configuration of the system at the beginning and end of a specified period.
            • interaction — When one object exerts a force that may change the state of motion (translational or rotational) of another object, those objects are said to interact.
            • internal force — A force exerted on one constituent of a specified system by another constituent of the same system. Internal forces do not affect the momentum of the system's center of mass, because their effects always cancel as required by Newton's Third Law.
            • kinematics — The branch of Newtonian mechanics that is the study and description of the possible motions of material bodies.
            • kinetic energy — The fundamental manifestation of mechanical energy, kinetic energy is the energy associated with an object's translational and/or rotational motion. Kinetic energy provides the definition of work (and hence all other forms of mechanical energy) through the Work-Kinetic Energy Theorem.
            • line of action — An infinite line passing through the point of application of a force parallel to the force vector.
            • magnitude — The length of a vector, or the absolute value of a scalar quantity. The magnitude is always a positive scalar value.
            • mass — A quantitative measure of the resistance of an object to attempts to change its velocity.   Hence mass is a measure of inertia (from the Latin vis inertia, the force of inaction.) 
            • massless object — An object that is treated as having no mass.{
            • mass on a spring — a common physics problem and the archetype of the system for simple harmonic motion
            • mechanical energy — The sum of the kinetic energy and any potential energies of a system.
            • mechanics — The study of the forces acting on material bodies and the resulting motions, if any.
            • model — In modeling physics a physical model describes the system, the state of its constituents (including perhaps geometric and temporal structure), their internal and external interactions, and has Laws of Change that determine the changes of state (i.e. behavior).  Models combine the definitions, concepts, procedures, interactions, laws of nature and other relationships that model some aspect of the physical world.  Models intermediate between laws of nature, which are relationships among abstract q
            • moment arm — Also called the "lever arm", the moment arm is the distance of closest approach between the line of action of a force and the axis of rotation. It is used to compute the torque produced by the force about the axis of rotation.
            • moment of inertia — A measure of the tendency of an object to maintain its rotational velocity about a specified axis of rotation.  The moment of inertia depends linearly on the mass and quadratically on the distance of that mass from the axis of rotation.  It plays the same role for rotational motion as mass plays for translational motion, being both the ratio of angular momentum to angular velocity and the ratio of torque to resultant angular acceleration, whereas mass is the ratio of (linear) momentum to velocit
            • momentum — Mass times velocity, or, equivalently, a quantity whose time rate of change is equal to the net force applied to a system.
            • motion diagram — A pictorial representation of the motion of an object. it usually takes the form of a one- or two-dimensional plot showing the position of the object at defined times.
            • natural frequency — The frequency that is characteristic of a given freely oscillating system, with no applied driving force.
            • net force — the vector sum of all forces acting on an item or system.
            • Newton's First Law — If an object is moving with no force acting upon it, then it will move with constant velocity. Note that velocity is a vector, so this statement implies that the object will keep the same speed and the same direction of motion.  This directly contradicts the animistic view of motion in which the natural condition of a body is at rest with respect to its surroundings - the First Law says the natural state of a body is moving with zero acceleration, not zero velocity.
            • Newton's Second Law — The mathematical relationship between force and momentum, or, for systems with constant mass, the relationship between force and acceleration.
            • Newton's Third Law — Every force exerted on one body by a second body is paired with another force of equal magnitude and opposite direction exerted on the second body by the first.
            • Newtonian mechanics — In principle, Newtonian Mechanics can be derived from only Newton's Laws (which include ∑F=ma) and the calculus of motion.
            • non-conservative force — A force which does work on an object in a path-dependent manner. For example, any force that has more than one possible value at a specific position is non-conservative.
            • parallel axis theorem — A relationship between the moment of inertia of a rigid body about an axis passing through the body's center of mass and the moment of inertia about any parallel axis.
            • pendulum — A pendulum is a physical object thatundergoes small angular oscillations under the restoring force of gravity.
            • period — The length of time it takes for a repeating motion to return to the same place
            • periodic motion — Motion which repeats after a fixed period of time, such as harmonic oscillation or orbital motion governed by a central inverse-square law force.
            • phase — A measure of the portion of the period that has passed since a given reference point in a case of periodic motion.
            • phenomenological forces — Macroscopic bodies are composed of huge numbers of elementary particles, which means that the effects of the fundamental forces on macroscopic bodies are complicated by the collective interactions of these particles. As a result, it is often advantageous to construct new force laws to describe the interactions of macroscopic bodies, even though these "new" forces are actually manifestations of the fundamental forces.
            • Physical Model — "A physical model (in physics) is a representation of structure in a physical system and/or its properties." [David Hestenes http://modeling.asu.edu].  A physical model will describe the system, the state of its constituents (including geometric and temporal structure), their internal interactions, external interactions, and the changes of state (that is to say, the system's patterns of behavior). 
            • point particle — An object that has no internal structure, and no physical size. Also commonly called a point mass.
            • position — A vector with dimensions of length giving the displacement of an object from the origin of a specified coordinate system.
            • position versus time graph — A plot of position as a function of time is an often useful diagrammatic representation of kinematics problems.
            • potential energy — A form of energy associated with the presence of conservative interactions such as gravity or a spring.
            • power — The time rate of doing Work.
            • precession — The slow rotation of the symmetry axis (which is also the axis of rotation) of a rapidly spinning symmetric object.
            • problem
            • projectile — An object that is hurled ( projected ) with force on a trajectory, but with no further force applied after that initial impulse.
            • pure rotation — A physical situation in which the only motion is rotation about a well-defined center of rotation, with no translational motion or storage of potential energy.
            • quantity of motion — Newton's archaic term for the quantity now known as momentum.
            • radians — A unit of angular measure. There are 2π radians in a full circle or a full rotation.
            • restoring force — A force directed opposite the displacement of a mass from some equilibrium position that acts to restore the mass to the equilibrium location. The most commonly analyzed case is a restoring force which has a magnitude linearly proportional to the displacement from equilibrium, leading to Simple Harmonic Motion.
            • right-handed coordinate system — a coordinate system consisting of three mutually perpendicular axes, x , y , and z , in which the z -axis follows the "right hand rule" as the direction of the cross product of axes x and y .
            • rigid body — An extended object which does not change shape.
            • rolling without slipping — A common assumption in problems of rotation, angular momentum, and torque, and one commonly encountered in reality. A rotating object, usually circular, rolls against another object (very often a flat surface) without any slipping between the rim of the object and the object or surface it is rolling against.
            • rotational kinetic energy — The kinetic energy associated with uniform rotation of a rigid body about an axis.
            • scalar — A quantity that does not have a direction associated with it.
            • scalar product — Also called the dot product, the scalar product is a special method of multiplying two vectors that gives as a result a scalar (that is, a quantity with magnitude but no direction).
            • significant figures — The non-zero individual digits ( figures ) in a number that are of importance and essential to convey the value and the precision needed or measured.
            • sinusoidal function — Sine, cosine or a linear combination of sine and cosine with arbitrary constant coefficients.
            • small angle approximation — when the angle is small, and expressed in radians, then we may approximate sin(θ) by θ.
            • speed — Speed is the magnitude of the velocity vector, and is therefore the scalar value corresponding to the velocity, regardless of direction (and always a positive quantity)
            • statics — The branch of mechanics which is the study of systems of forces acting on bodies which are not in motion.
            • Strategic Knowedge
            • system — The object or the group of objects whose motion is being described using a model.
            • system constituent — A distinct object within the system being considered.
            • tangential acceleration — The component of an object's total acceleration directed tangential to the path of the object's motion. The tangential component of the acceleration changes the object's speed, but does not affect the object's direction of motion.
            • tension — The force exerted by a string, rope, tape, or other flexible object.
            • torque (single-axis) — An interaction which has the potential to produce a change in the rotational velocity of a system about a specified axis.
            • totally inelastic collision — During the course of the collision the colliding objects become attached to form a single rigid body. (Also often called a perfectly or a completely inelastic collision.)
            • vector — A physical quantity that has both magnitude and direction.
            • velocity — The time rate of change of position.
            • weight — The force of gravity on an object near the earth's surface (or the surface of some other planet).
            • work — An interaction which produces a change in the mechanical energy of a system, or the integrated scalar product of force and displacement.
            • Work-Kinetic Energy Theorem — The relationship between the kinetic energy of a point particle and the work done on the point particle. This theorem is one way to arrive at a mathematical definition of work.

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            IV.  Instructional Paths

            Sequences of lessons that suggest an order in which to learn the material and fill in details. This organizational scheme for the WIKI is designed to help beginners get started learning physics.
            • Learning Modules — Not a single organized path, but rather a collection of the "footsteps" along the Instructional Paths, Learning Modules give instruction related to a single topic.
            • MIT 8.01 Lessons (Pritchard) — This sequence of lessons is closely based on the classes of Prof. David E. Pritchard in Introductory Physics Course 8.01 at MIT.  This organizational scheme is designed to help students learn the Modeling Applied to Problem Solving pedagogical approach of Prof. Pritchard and his RELATE educational group (particularly Dr. Andrew Pawl and Dr. Annalia Barrantes).

            V. Worked Examples

            Another way experts commonly organize the mechanics syllabus is by the tasks that students should learn to perform.

                Worked Examples from Motion and Acceleration

                Worked Examples from Momentum and Force

                Worked Examples from Mechanical Energy and Non-Conservative Work

                Worked Examples from Angular Momentum and Torque

                Multi-Concept Problems

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                  Worked Examples from Motion and Acceleration

                  • Accelerate, Decelerate — Determining the relationships between position, velocity and acceleration from a position vs. time plot.
                  • An Exercise in Continuity — An introduction to continuously piecing together kinematic solutions for time intervals with different accelerations.
                  • A Velocity for Words — Put the motion described by these graphs into words.
                  • Campus Tour — Basic problem to illustrate graphical representation of position and velocity.
                  • Dwarf Mistletoe — Perhaps this parasitic plant should be called "Dwarf Missiletoe".
                  • Fan-Powered Ice Boat — Graphing 1D motion with constant acceleration.
                  • Fountain — Modern fountains are an excellent example of projectile motion.
                  • Keys Please — Keys moving in 1D freefall with or without initial velocity.
                  • Lissajous Figures and the Bowditch Pendulum — Image generated by a Pendulum with two natural Frequencies.
                  • Overdriving Headlights — How long can you drive at constant velocity before you have to hit the brakes, assuming standard night detection distances?
                  • Space Station — find the approximate magnitude of the acceleration experienced by the space station as a result of the gravitational pull of the earth.
                  • Speed Trap — Police car with constant acceleration must catch speeder with constant velocity.
                  • Throwing a Baseball 1 (The Basics) — How far will the ball travel horizontally from the instant it leaves your hand until the instant it first contacts the ground?
                  • Throwing a Baseball 2 (Vectors) — How fast is the ball moving just before it impacts the ground?
                  • Training Flight — In this example we will calculate acceleration, time, speed, and distance assuming constant acceleration.
                  • Where Do We Meet? — Two people moving in one dimension with constant speed are destined to meet – but where?

                  Worked Examples from Momentum and Force

                  • Apparently I've Lost Weight — Finding apparent weight using normal force.
                  • Atwood's Machine — The standard pulley problem as an example of systems.
                  • Basics of Static Friction — An introduction to determining the size of the static friction force.
                  • Cannonballs in a Boxcar — Examining the concept of Center of Mass and Conservation of Momentum in Different Ways.
                  • Chain Reaction — A series of elastic collisions.
                  • Finding Normal — Several examples showing how to find the normal force in common situations.
                  • Head-on Collision — Compare the forces on the occupants of two cars in a 1-D totally inelastic collision.
                  • I'm Inclined to Tilt the Coordinates — Basic inclined plane problems illustrating the advantage of tilted coordinates.
                  • Is That Normal? — Several examples illustrating how to find the normal force in not-so-common situations.
                  • Let it Rain — Analyzing a continuous momentum flux (falling water).
                  • Momentum Transport — Analyzing a continuous momentum flux (water from a hose).
                  • Off the Wall — Simple problem illustrating the definition of impulse and the utility of an initial-state final-state diagram.
                  • Out of Bounds — A typical perfectly inelastic collision in 2-D.
                  • Pushing a Box — A person pushes a box of mass 15 kg along a smooth floor by applying a perfectly horizontal force F.
                  • Pushing a Box Some More — A person pushes a box of mass 15 kg along a smooth floor by applying a force F at an angle of 30° below the horizontal.
                  • Pushing a Box with Friction — Assuming the coefficient of kinetic friction between the box and the ground is 0.45, what is the magnitude of F?
                  • Pushing a Box with Friction Some More — A person pushes a box of mass 15 kg along a floor by applying a force F at an angle of 30° below the horizontal. There is friction between the box and the floor
                  • Pushing Two Boxes — A person pushes a box of mass 15 kg along a smooth floor by applying a perfectly horizontal force F. In the process, the 15 kg box pushes against another box with a mass of 10 kg and causes it to move.
                  • Rope Bridge — The tension in ropes supporting an object can sometimes be much larger than the object's weight.
                  • Skydiving — Explore the force from air resistance acting on a skydiver at various stages of the dive.

                  Worked Examples from Mechanical Energy and Non-Conservative Work

                  Worked Examples from Angular Momentum and Torque

                  Multi-Concept Problems

                  • Accelerating Flywheel — Acceleration of a symmetric object about a fixed axis under constant torque (single-axis).
                  • A Walk on the Pond — How far will two children slide after a perfectly inelastic collision?
                  • Banking the Curve — Two examples of drawing free body diagrams for objects navigating a banked curve.
                  • Big Ben — Check Parliament's math by calculating the period of Big Ben's pendulum.
                  • Capture Cross-Section — Calculation of Effective Cross-Section of a planet with gravity (interaction)
                  • Close the Gate — Classic example of static friction on a moving surface.
                  • Cue the Right-Angle Bracket — Learn a valuable shortcut for dealing with a specific kind of elastic collision.
                  • Down the Alley — Determine the final speed of a ball that is initially sliding without rotation.
                  • Down the Ramp — Find the acceleration of a ball rolling without slipping on an inclined plane.
                  • Down the Well — A mass falling while attached to a massive pulley.
                  • Locking Your Bike — Determine how the weight of a bicycle plus rider is divided between the wheels in various circumstances.
                  • Mass Between Two Springs — A case of Simple Harmonic Motion.
                  • Mass on a Vertical Spring — Another case of Simple Harmonic Motion, this time with gravity (near-earth) thrown in.
                  • Pull Harder! — Examine the work needed to change the radius of rotation of an object rotating in a circle.
                  • Rotating a Space Ship — Changes in Angular Velocity when the Moment of Inertia is changed, but no torque applied.
                  • Sliding Yardstick — What happens to a yardstick (or meter stick) supported by two fingers as those fingers are slowly moved toward each other?
                  • The Ladder Problem — A standard statics problem.
                  • Watch Your Head — Consider the impulse and average force delivered to the head of a player performing a "header" in soccer.

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