Scientific Notation, Units and Uncertainty:

• *Scientific notation review. [2021.08.14]
• *SI prefixes [2021.08.16]
• *Units of volume and conversions [2021.08.16]
• *Example:  unit conversion (speed) [2021.08.17]
• *Example: unit conversion (density).[2021.08.17]
• *Example:  unit conversion (seconds in a year) [2021.08.17]
• *Example:  unit conversion (miles in a light year) [2021.08.17]
• *Significant digits:  addition and subtraction [2021.08.18]
• *Significant digits:  multiplication and division [2021.08.18]
• *Propagation of error (important for lab 1 calculations!) [2021.08.19]

Introduction to Vectors:

• *Introduction to vectors and geometric vector addition. [2021.08.21]
• *Vector components introduction and examples. [2021.08.21]
• *Quantitative vector addition using components [2021.08.22] 
• *Vector algebra:  scalar multiples, unit vectors, components [2021.08.23]
• *Vector algebra:  vector opposites, zero vector, vector subtraction [2021.08.23]
• *Vector algebra:  dot product, geometric and algebraic views [2021.08.23]

Additional Examples (previous quiz and test items):

• *Example:  number of football fields to cover the Earth [2023.11.02]
• *Example:  magnitude of a vector difference, angle between two vectors [2023.08.31]
• *Example:  number of football fields along the equator [2023.08.25]
• *Example:  moment of inertia error propagation problem [2022.09.27]
• *Example:  propagation of errors with spring potential energy formula. [2022.08.28]
• *Example:  unit conversion and significant digits (mass of a free proton) [2022.06.16]
• *Example:  addition of force vectors by components [2022.03.23]
• *Example:  volume of a cylinder/propagation of error [2022.03.21]
• *Example:  density of the moon [2022.03.21]
• *Example:  area of a rectangular property using propagation [2022.02.27]
• *Example:  atoms of gold in one ounce [2022.02.07]
• *Example:  propagation of error with centripetal acceleration formula. [2021.10.14]
• *Example:  compute the vector difference in polar form. [2021.09.17]
• *Example:  propagation of error with kinetic energy formula [2021.09.17]

Introduction to 1D Kinematics:

• *Position time graphs and slope, velocity time graphs and area. [2021.08.27]
• *Velocity time graphs and slope, acceleration time graphs and area. [2021.08.27]
• *Derivative and integral relations between position, velocity and acceleration + examples. [2021.08.29]
• *Deriving the equations for constant acceleration kinematics + examples. [2021.08.29]
• *Measuring speed with an iPhone. [2021.02.27]
•  Common mistakes in physics:  area bounded by a velocity time graph [2023.08.29]
• *Short animation:  animating motion on an x-t graph and in a 1D space simultaneously [2023.09.01]
• *Short animation:  animating motion on a v-t graph and in a 1D space simultaneously [2023.09.01]

Additional Examples (previous quiz and test items):

• *Example:  free fall on the moon, feather and hammer experiment [2023.09.29]
• *Example:  minimum acceleration to avoid a collision [2023.08.29]
• *Example:  equations of motion for a linearly increasing acceleration. [2023.08.27]
• *Example:  vertical free fall from rest, flight time and final speed [2023.08.25]
• *Example:  1D velocity function given, compute acceleration and displacement [2022.08.30]
• *Example:  velocity data, estimate the displacement and acceleration [2022.08.29]
• *Example: car accelerates then brakes to a stop, find the max speed and total displacement. [2022.06.27]
• *Example:  v-t graph, make an a-t graph and compute average velocity [2022.06.24]
• *Example: vertical free-fall upward launch, height and speed [2022.03.22]
• *Example:  v-t graph, displacement and average velocity [2022.03.22]
• *Example (pebble launched downward from a bridge) [2022.02.07]
• *Example (sinusoidal position-time graph: speed and velocity) [2022.02.07]
• *Example:  car brakes and then coasts, find the total displacement. [2021.10.14]
• **Example:  vertical free fall - time and height at which two rocks pass. [2021.09.17]
• *Example:  cart accelerates down a ramp, final speed and distance given. [2021.09.17]
• *Example:  velocity and average velocity from a v-t graph [2021.09.17]
• *Example:  position and time for two cars to pass. [2021.04.24]
• *Example:  displacement and average velocity from a v-t graph. [2021.04.24]
• *Example:  two cars crash. [2021.03.06]
• **Example:  vertical free fall + time to hear the crash. [2021.03.06]
• *Example:  acceleration of a bullet hitting a tree. [2021.03.06]
• *Example:  car slowing to a stop. [2021.03.06]
• *Example:  displacement vs. distance, average velocity vs. speed. [2021.03.06]

Kinematics in 2D:

• *2D Kinematics:  position, velocity and acceleration vectors in higher dimensions. [2021.09.05]
• *How the acceleration vector affects the velocity vector. [2021.09.05]

Projectile Motion:

• *Derivation of the equations of motion for projectile motion + example with flight time and range. [2021.09.05]
• *Animation and description of vectors in projectile motion [2023.09.22]
• *Short animation of vectors in projectile motion [2023.09.22]
• *How to compute the impact velocity of a projectile [TBA]
• *Proving that projectile motion is parabolic [TBA]
• *Example: Projectile motion with object landing higher than the launch. [2021.03.08]
• *Example: Projectile motion with a horizontal launch from a height (related to this week's lab). [2021.03.07]
• *Example: Projectile flight terminates at a known value of x, and compute the impact velocity. [2021.03.08]

Introduction to Circular Motion:

• *Angular speed and angle as a function of time for uniform circular motion [2022.08.30]
• *The relationship between linear and angular velocity [2022.08.30]
• *Short:  converting RPM to radians/second. [2022.09.06]
• *Equations of motion for an object in uniform circular motion + derivation of centripetal acceleration formula v^2/r [2022.08.30]
• *Animation:  position vector, velocity vector and acceleration vector for uniform circular motion [2022.08.31]
• *Example:  compute the centripetal acceleration and tangential velocity at the edge of a disk. [2020.10.05]
• *Example:  rotations per minute to radians per second, tangential velocity, centripetal acceleration. [2020.10.07]

Relative Motion:

• *Relative velocity and two examples [2022.09.01]

Additional Examples (previous quiz and test items):

• *Example:  accelerated circular motion find the position and velocity vectors [2023.10.09]
• *Example:  compute the launch angle and max height for a baseball throw [2023.10.06]
• *Example:  compute the launch angle so impact happens at the top of the trajectory [2023.09.15]
• *Example:  projectile lands on a roof, time, range, impact speed [2022.10.03]
• *Example:  projectile launched horizontally, use range to find the speed [2022.09.23]
• *Example:  Projectile launched across a gorge, time for impact and drop in height before impact. [2022.05.27]
• *Example: projectile motion launch from a cliff: max height, flight time, impact angle [2022.03.23]
• *Example (projectile with horizontal launch, unknown height) [2022.02.17]
• *Example:  Another projectile launched across a gorge, time, final height, final y-velocity and impact velocity at known x coordinate. [2021.10.15]
• *Example:  projectile launched downward at an angle from a cliff. [2021.09.27]
• *Example:  velocity vector given as a function of time in 2D, find the position vector and average velocity vector [2021.09.27]
• *Example: Projectile with tilted launch from a cliff. [2021.04.25]
  
• *Example:  projectile motion on a level surface, flight time, range, max height [2021.03.23]

Newton's First Law (N1):

• Introduction to Newton's First Law [TBA]
• Application of Newton's First Law [TBA]

Newton's Second Law (N2):

• *Introduction to Newton's second law:  why F=ma + two examples [2022.09.03]
• *Mass, weight and normal force + three examples [2022.09.04]
• *Normal force on a tilted surface, components of the weight vector + inclined plane example [2022.09.05]
• *Short animation:  a trick for finding vector components on an incline [2022.09.06]
• *Example:  mass suspended from three strings [2022.09.08]
• *Example: horizontal Atwood machine find the tension and acceleration [2023.09.04]
• *Example:  normal force on an elevator ride [2022.09.25]
• *Common mistake: vector magnitudes don't add! [2023.09.09]
• *Common mistake: five examples where normal force is not mg. [2023.09.10]
• *Common mistake: T is not mg in an accelerating Atwood machine [2023.09.10]

Newton's Third Law (N3) and Review of N1-N3:

• Newton's Third Law Intro [TBA]
• *Quick summary of Newton's Laws (old video)

Additional Examples (previous quiz and test items)

• *Example:  pulling up at an angle on a block, angle that maximizes acceleration [2023.10.07]
• *Example:  two blocks accelerated vertically, given the breaking strength of the strings [2023.09.30]
• *Example:  pushing three blocks on a smooth surface [2023.09.16]
• *Example: inclined Atwood machine with M and 2M [2022.03.24]
• *Example (scale on an elevator stationary vs. accelerating) [2022.02.17]
• *Cumulative Example:  force vector as a function of time given in 2D, compute r(t) and the average velocity vector. [2021.10.18]
• *Example:  pushing on a block that pushes another block, given acceleration, compute applied and interaction forces. [2021.09.27]
• *Example:  change in velocity and position from a constant force-time graph. [2021.06.14]
• *Example:  pushing a combination of blocks. [2021.04.25]
• *Example:  acceleration and braking distance given the braking force [2021.03.23]
• *Example:  Horizontal Atwood Machine (old video) [2020.09.11]
• *Example:  Tilted Atwood Machine (old video) [2020.09.11]
• *Example:  a block sitting on a block. (old video) [2020.09.15]
• *Example: an applied force with a downward component. (old video) [2020.09.15]
• *Example:  level surface, rope pulling up at an angle. (old video) [2020.09.25]

Miscellaneous Dynamics Problems

• *Introduction to equilibrium. (old video) [2020.09.22]
• *Example:  mass suspended from three strings [2022.09.08]
• *Example:  sliding down an incline with constant velocity. (old video) [2020.09.22]
• *Example:  angle of tilt for fuzzy dice in an accelerating car. (old video) [2020.09.22]

Problems with Static and Kinetic Friction

• Introduction to static and kinetic friction [TBA]
• *Example:  two blocks with friction, sliding at constant velocity, find the coefficient of kinetic friction. [2021.06.16]
• *Example:  breakaway force and acceleration of a block with friction. [2020.09.22]
• *Example:  breakaway force when the force is pulling up at an angle. [2020.09.22]
• *Example [related to lab 5]:  breakaway force. [2020.09.22]
• *Example [related to lab 5]:  breakaway angle for a block on a ramp. [2020.09.22]
• *Example [related to lab 5}:  sliding to a stop. [2020.09.22]
• *Example [related to lab 5]:  block sliding on an inclined plane with friction. [2020.09.22]

Uniform Circular Motion:

• *Review:  uniform circular motion kinematics and derivation of centripetal acceleration formula [2022.08.30]
• *Introduction to centripetal force + the simplest UCM problem. [2020.09.24]
• *Example:  UCM on a vertical path. [2020.10.22]
• *Example: find the minimum speed to make it through a loop. [2020.10.11]
• *Example:  UCM at an angle (conical pendulum) [2020.09.23]
• *Example:  measure the mass of a stopper using UCM. [2020.10.07]
• *Example:  car rounding a curve, relying on friction. [2020.09.24]
• *Example:  banked turn, perfect speed and angle to avoid friction. [2020.09.24]

Additional Examples (previous quiz and test items):

• *Example:  heavy rope with friction find the tension as a function of position [2023.10.09]
• *Example:  tilted Atwood machine with friction, minimum mass to break loose [2023.10.07]
• *Example:  given velocity time data for a slide to a stop, find coefficient of friction [2023.10.02]
• *Example:  maximum static friction force so the string won't break [2023.09.18]
• *Example:  rotating swing ride problem [2022.11.23]
• *Example:  block pulled under another block at constant speed [2022.10.11]
• *Example:  sliding up and down the ramp, total travel time and more [2022.10.06]
• *Example:  kinetic friction speeds up a sliding box [2022.06.13]
• *Example:  four blocks dragged at constant speed with kinetic friction, find all the tensions. [2022.05.30]
• *Example: Breaking strength of a string in UCM. [2022.04.02]
• *Example: block dragged by a rope, including friction [2022.03.24]
• *Example: artificial gravity using the rotation of a space station. [2021.11.24]
• *Example:  block sliding down a ramp with given acceleration and angle, compute the friction force, coefficient, etc. [2021.10.15]
• *Example:  amusement park ride with a swing spinning in circles attached to cables. [2021.10.18]
• *Example:  Atwood machine with a block on top of the cart, held in place with static friction. [2021.10.15]
• *Example:  vertical Atwood machine, acceleration and time to hit the ground. [2021.09.27]
• *Example:  Atwood machine sliding at constant speed due to a friction force. [2021.09.27]
• *Example:  Tilted Atwood machine with friction. [2021.04.25]
• *Example:  apparent weight at the equator. [2021.04.25]
• *Example:  Atwood machine with static and kinetic friction. [2021.04.07]
• *Example:  car rounding a corner, find the angle of the fuzzy dice. [2020.11.25]

Note: the only new videos for this module are on the basics of gravitation and circular orbits, but in-class problem solving has advanced dynamics problems.

Short intro to gravitation and circular orbits videos:  

• *Introduction to Newton's Law of Gravitation, the Cavendish experiment, and an example. [2021.10.06]
• *Example:  orbital period and speed of the LRO lunar orbiter probe. [2022.04.03]
• *Example:  weighing the Sun using Jupiter's orbit [2022.10.15]

Note: most past exam material (2020 - 2023) has been folded into the "additional examples" videos in the prior modules. Actual exams and video solutions (2024 - present) will be provided here [TBA]

Work and Kinetic Energy

• *Definition and units of work:  constant force and direction case (no calculus) [2020.10.06]
• *Why is work defined this way?  Path independence of gravity example. [2020.10.06]
• *Forces of constraint do no work. [2020.10.06]
• *Example:  calculate work using a force vs. displacement graph. [2020.10.08]
• *Example:  work done by applied force and friction for a sliding block. [2020.11.25]
• *Example:  final velocity and displacement from constant force-displacement graph. [2022.05.26]

Applying the Integral Definition of Work/Hooke's Law:

• *Definition of work:  general definition as an integral in 1 dimension + an example. [2020.10.06]
• *Where does the formula for kinetic energy come from?  Derivation of the work-energy theorem. [2020.10.06]
• *Example:  net work and kinetic energy for a sliding block. [2020.11.25]
• Force in Hooke's Law springs [TBA]
• *Example:  work done by a Hooke's Law spring vs. work done on a Hooke's Law spring. [2020.10.06]
• *Completely general formula for work. [2020.10.06]
• *Example:  work done along a curved path. [2020.10.06]
• *Example:  work done by gravity as an asteroid falls to Earth [2023.10.29]

Introduction to Power:

• *Introduction to power, examples, power vs. time graphs, constant velocity formula. [2021.10.05]

More Examples:

• *Example:  final speed of a block pushed down at an angle on a level surface with friction [2023.10.26]
• *Example:  non-linear force function, compute the total work [2022.10.17]
• *Example:  block pulled at an angle with a rope including friction, final speed using work/energy. [2022.06.29]
• *Example:  block sliding down a ramp with friction, distance and final speed are given, find friction force etc using work. [2021.11.22]
• *Example:  net work and kinetic energy for a sliding block. [2020.11.05]
• *Example:  work done by tensions in a sliding block problem. [2020.10.22]
• *Example:  sliding block with force pulling at an angle, calculate work including friction. [2020.10.08]
• *Example:  block sliding down a ramp at constant speed, work done by friction. [2020.10.22]

Conservative Forces, Potential Energy and Conservation of Energy

Theory Section

• *Review:  Where does the formula for kinetic energy come from?  Derivation of the work-energy theorem.
• *Potential energy and the work done by gravity.
• *Potential energy and the work done by a spring.
• *Quick proof of conservation of energy, with and without friction.
• Force in Hooke's Law springs. (third party video)
• *Review:  work done by a Hooke's Law spring vs. work done on a Hooke's Law spring
• *Finding force and equilibrium position from a potential energy function.

Additional Examples (previous quiz and test items)

• *Example:  Mass dropped from rest while attached to a spring. [2024.05.20]
• *Example:  Atwood machine with three masses, work/energy approach.[2023.12.18]
• *Example:  energy released in an asteroid impact by directly computing the work done by gravity [2023.10.29]
• *Example:  maximum height of a block shot upward by a spring [2023.10.27]
• *Example: block shot up a ramp by a spring [2022.11.24]
• *Example:  spring launches a block that later slides to a stop on a rough surface [2022.10.15]
• *Example:  block bounces off a spring, max compression and max acceleration. [2022.08.05]
• *Example:  another speed at the bottom and tension in the string for a pendulum. [2022.05.31]
• *Cumulative Example:  maximum height before pendulum string breaks. [2022.05.06]
• *Example:  maximum compression of a spring (symbolic) with and without friction. [2022.05.08]
• *Example: Spring gun, potential energy, spring constant and average power output. [2022.04.03]
• *Cumulative Example:  cart rolling into a hemispherical bowl - find the speed and normal force at the bottom. [2021.11.21]
• *Example:  investigating the Hooke's law potential to verify the stable equilibrium position. [2021.10.30]
• *Cumulative Example:  giant spring shoots a bowling ball into projectile motion. [2021.10.30]
• *Cumulative Example:  pendulum speed at the bottom and tension at the bottom. [2021.06.17]
• *Example:  block sliding down a ramp into a spring. [2021.04.30]
• *Cumulative Example:  uniform circular motion with a spring. [2021.04.07]
• *Example:  roller coaster through the loop (much greater than the minimum speed). [2020.10.22]
• *Example:  roller coaster through the loop (minimum height to make it through the loop). [2020.10.15]
• *Review Example: find the minimum speed to make it through a loop. [2020.10.11]
• *Example:  car with a springy bumper, find the maximum compression and acceleration. [2020.11.25]
• *Example:  ball dropped on a spring, find the spring constant. [2020.10.22]
• *Example:  spring shooting a mass horizontally. [2020.10.11]
• *Example:  three projectiles launched with the same initial height and speed. [2020.10.11]

Momentum and Collision Problems:

• *Definition of momentum and impulse + a quick example. [2020.10.19]
• *Momentum and its relation to force + average force example. [2020.10.19]
• *Example:  impulse from a force-time graph.  Find the final speed and the work done. [2020.10.19]
• *Why is momentum conserved?  Quick derivation starting from Newton's third law. [2020.10.19]
• *Example:  perfectly inelastic collisions and illustration that energy must be lost. [2020.10.19]
• *Example:  perfectly elastic collisions. [2020.10.19]
• *Example:  2D inelastic collision. [2020.10.23]
• *Example:  2D elastic collision. [2020.10.23]

Center of Mass and the COM Reference Frame:

• *Center of mass in 1D and a simple example. [2020.10.26]
• *Center of mass in 2D and a simple example. [2020.10.26]
• *Center of mass velocity and the center of mass reference frame. [2020.10.26]
• *Example:  1-D collision analyzed in the center of mass reference frame. [2020.10.26]
• Center of Mass Integrals [TBA]

Additional Examples:

• *Example:  Elastic collision in the center of mass frame. [2024.05.13]
• *Example:  three particles net momentum, cm velocity, cm frame velocities and net momentum [2022.12.28]
• *Example: dart impacts a block connected to a spring [2022.12.01]
• *Example: general solution for an elastic collision with a stationary target [2022.11.26]
• *Example:  two masses M and 2M spring apart with energy E. [2022.11.01]
• **Cumulative Example:  sliding blocks spring apart, find final speed and initial spring energy. [2022.08.10]
• *Example: 1-D elastic collision in the CM frame. [2022.06.08]
• **Cumulative Example: inelastic collision with friction before and after the collision. [2022.06.01]
• *Cumulative Example:  pendulum partially elastic collision. [2022.05.10]
• *Cumulative Example:  fan cart work, power, impulse, average force. [2022.05.09]
• *Example: clay blob used to stop a block [2022.04.21]
• **Example:  mass explodes into three pieces, compute all the speeds. [2021.11.23]
• *Example:  hay bail falls vertically into a cart moving  horizontally. [2021.11.08]
• *Example:  multiple inelastic collisions. [2021.06.19]
• *Example:  final velocities for masses pushed apart by a coiled spring. [2021.06.18]
• *Example:  rubber ball shot at a cart. [2021.05.01]
• **Example:  velocity after n inelastic collisions. [2020.11.27]
• **Cumulative Example:  the ballistic pendulum. [2020.11.25]
• **Example:  compute the maximum compression of a spring during a collision. [2020.10.23]
• *Example:  an explosion with known energy -- what percent of energy goes into each piece? [2020.10.22]

Rotational Kinematics and Rotational Energy

• *Rotational variables and constant angular acceleration equations + example. [2020.11.02]
• *Relating linear and angular speed and acceleration + example. [2020.11.02]
• *Example:  a grinding wheel slowing to a stop. [2020.11.02]
• *Rotational energy and moment of inertia derivation + example. [2020.11.02]
• *Example:  rotational kinetic energy of the Earth. [2020.11.02]
• *Example:  conservation of energy -- disk with a falling mass. [2020.11.02]
• *Example:  conservation of energy -- horizontal Atwood machine with a heavy pulley. [2020.11.02]
• *Example: conservation of energy -- vertical Atwood machine with a heavy pulley. [2020.11.05]
• *Example:  work and energy -- spinning up a disk using a string wrapped around the hub (related to lab) [2020.11.02]

Torque and Angular Acceleration

• *Defining torque + simple examples. [2020.11.03]
• *Quick derivation of rotational N2 (net torque causes angular acceleration) [TBA]
• *Motivation for rotational N2 + simple example:  torque causes angular acceleration. [2020.11.03]
• *Example:  same disk/hub from work/energy example, but solve it using torque and angular acceleration (related to lab) [2020.11.03]
• *Example:  previous heavy disk with mass pulling down, solved using forces and torques [2020.11.08]
• *Example:  previous vertical Atwood with heavy pulley, solved using forces and torques [2020.11.08]
• *Cumulative Example:  grinding wheel slowed by friction. [2020.11.03]
• *Example:  another disk accelerated by pulling a string/hub:  torque, angular acceleration, kinetic energy (related to lab) [2020.11.25]

Additional Examples (previous quiz and test items):

• *Example:  Basketball spinning to a stop. [2024.05.27]
• *Example: hollow ball rolls without slipping down a ramp [2022.11.28]
• *Example:  Atwood with M, 2M and heavy disk pulley. [2022.11.02]
• *Cumulative Example:  ramp atwood - unwinding a heavy pulley. [2022.05.17]
• *Example:  disk accelerated by a motor:  final speed, average acceleration and torque [2022.04.21]
• **Example: two pulleys with two hanging masses going opposite directions (force and torque analysis). [2022.06.09]
• *Example:  average torque and power output of a motor given final angular velocity of the heavy disk. [2022.06.02]
• *Example:  torque exerted by friction and work done by friction as a disk slows to a stop. [2021.06.20]
• *Example:  given MOI, torque and time for a ceiling fan, find angular acceleration, final angular speed and power. [2021.11.08]

Torque Due to Gravity + Gravitational PE for a Rigid Body

• *Gravitational potential energy for a rigid body + example. [2020.11.03]
• *Gravitational potential energy for a rigid body + example, new version, new example! [2021.11.03]
• *Torque as a cross product and derivation of torque exerted by gravity on a rigid body. [2021.11.03]
• *Example:  torque exerted by gravity on a horizontal rod attached to a hinge. [2020.11.03]

Combined Translation and Rotation

• *Derivation of total energy for a rotating, translating object. [TBA]
• *Statement of total energy of an object with both translation and rotation (no derivation) + simple example. [2020.11.06]
• *Condition for rolling without slipping + application to a rolling ball (% energy) [2020.11.06]
• *Example:  ball vs. disk rolling down a hill, energy approach [2020.11.06]
• *Derivation of net force/center of mass acceleration formula + example (falling chain) [2021.11.06]
• *Derivation of the net torque/angular acceleration about the CM formula [TBA]
• *General dynamics:  net force and net torque + a simple example (unwinding disk) [2021.11.04]

Angular Momentum Lite (precession and derivation of theorems skipped for now)

• *Angular momentum and conservation for pure rotation + simple example (kid on a merry-go-round) [2020.11.06]
• *Example:  revolving mass on a table top pulled to a smaller radius [2020.11.07]
• *Angular momentum for a particle moving with a translational velocity + example with a particle impact. [2023.11.06]
• *Precession of a rotating object - derivation and demonstration [TBA]

Additional Examples (previous quiz and test items):

• *Example:  Thin rod ballistic pendulum. [2024.07.22]
• *Example:  Rotational Atwood machine, force/torque analysis. [2024.01.01]
• *Example:  Solid ball rolling up a hill, force/torque analysis. [2023.12.26]
• *Example: rod hinged at the bottom falling to the ground [2022.11.27] 
• *Example:  ring rolling down an incline. [2022.05.16]
• *Example:  retracting masses on a rotating disk. [2022.05.11]
• **Example:  disk initially launched on a surface with a backwards spin, find the distance until it rolls without slipping. [2021.11.26]
• *Example:  disk rolling down an incline on an axle with a small radius [2021.11.25]
• *Example:  angular momentum conservation:  first approximation of white dwarf spin rate [2020.11.07]

Static Equilibrium

• *Conditions for static equilibrium + 2 simple examples (point mass vs. extended object suspended from strings) [2020.11.08]
• *Example from canceled lab:  mass suspended from strings + law of cosines [2020.11.08]
• *Example from canceled lab:  balancing point for a rod and mass system [2020.11.08]
• *Translating a force vector to compute torque (useful short cut!) [2020.11.18]
• *Example:  cat on a table, compute forces on the legs. [2020.11.18]
• *Example:  horizontal beam:  tension and force exerted by the hinge. [2020.11.18]
• *Example:  angled beam:  tension and force exerted by the hinge. [2020.11.18]
• *Example:  beam supported by string with one end resting on smooth floor. [2020.11.20]
• *Example: ladder resting against a wall with friction on the floor, minimum angle to not slip. [2020.11.20]

Additional Examples (previous quiz and test items)

• *Example:  General solution of the slipping ladder (max height of the person). [2024.01.08]
• *Example: person walks on a beam suspended by cables [2022.12.11]
• *Example:  horizontal beam with tilted cable and additional suspended mass. [2022.08.12]
• *Example:  horizontal rod resting on a hinge and a slope [2021.12.09]
• *Example:  two beams propped against each other, supported by a rope between them. [2020.11.20]
• *Example:  horizontal beam hinged at the center [2020.11.25]

Note: most past exam material (2020 - 2023) has been folded into the "additional examples" videos in the prior modules. Actual exams and video solutions (2024 - present) will be provided here [TBA]

Fluid Statics

• *Density + examples.
• *Pressure as force per unit area + examples.
• *Quick note on absolute vs. gauge pressure.
• *Pressure at depth in a fluid + example.
• *Example:  pressure from a water tower.
• *Example:  open-tube manometer.
• *Example:  how a hydraulic jack works.
• *Example:  how a siphon works.
• *Buoyancy:  Archimedes' principle and two examples (floating glacier ice, apparent weight of a basalt cobble).

Fluid Dynamics

• *Continuity equation for a nearly incompressible fluid.
• *Pressure as energy density + microscopic view of what's going on.
• *Bernoulli's equation.
• *Example:  water tower hydrostatic pressure and water velocity at a faucet.
• *Example:  water flow meter based on Bernoulli's equation.

Additional Examples (previous quiz and test items)

• *Example: volume of hot air required to lift a hot air balloon [2022.12.22]
• *Example: density of a submerged rock.
• *Example:  minimum gauge pressure magnitude required to draw bubbles from a submerged tube.
• *Example:  another venturi flow meter, using open manometers for the pressure.
• *Example:  continuity equation with a big pipe splitting into two smaller pipes
• *Example:  water column with mass on top, pressure and fluid speed for a leak

Simple Harmonic Oscillator

• *Simple harmonic oscillator introduction and example.
• *Derivation of equations of motion for the simple harmonic oscillator.
• *The phase angle form of the general solution + example.
• *Example:  simple SHO example
• *Phase space diagrams for the simple harmonic oscillator.

Pendulums

• *Simple pendulum introduction
• *Expanding the sine function in a power series (important for the derivation below).
• *Derivation of equations of motion for the simple pendulum.
• *Example:  simple pendulum example.
• *The physical pendulum:  derivation of equations of motion and example.

Additional Examples (previous quiz and test items)

• *Example: spring constant and period for a mass hanging on a spring.
• *Example:  amplitude and maximum speed for a harmonic oscillator.
• *Example:  harmonic oscillator with inelastic collision.
• *Example:  spring constant and frequency given mass and the initial force and stretch .
• *Cumulative Example:  block bounces off a spring, time for the bounce and average force

Moment of Inertia and MOI integrals:

• *Review:  derivation of moment of inertia for a distribution of point masses. [old video]  
• *Example:  computing the moment of inertia for a rod rotated about one end. [old video]
• *Example:  computing the moment of inertia for a thin ring and thin cylindrical shell. [2023.01.09]
• *Example:  computing the moment of inertia for a disk and solid cylinder [2023.01.16]
• *Example:  moment of inertia for a hollow sphere using rings. [2023.08.23]
• *Example:  moment of inertia for a right triangular slab, axis colinear with one leg. [2024.08.27]
• **Example:  moment of inertia for a rod with variable density lambda =kx. [2024.08.31]
• **Example:  moment of inertia for a solid ball with variable density rho=kr. [2024.09.01]

The Parallel Axis Theorem and Review of the Physical Pendulum:

• *Derivation of the parallel axis theorem + three examples of using the PAT [2023.10.20]
• *Derivation of the period of oscillation for the physical pendulum + example [2024.10.06

*Note: this material may be taught in first or second semester physics, depending on the school.

Waves and Waves on a String

• *Frequency, wavelength, amplitude and wave speed.
• *Tension, linear density and wave speed on a string + example.
• *Fixed and free reflections, superposition principle., constructive/destructive interference.
• *Resonance for a string clamped at both ends + driving at a resonant frequency.
• *Example:  another resonance example for a string - many frequencies superposed.
• *Example:  resonance on a string, find the 2nd harmonic based on the 3rd harmonic.
• *Example:  wave speed and tension given the linear density and frequency of the 5th harmonic.
• *Example:  given two normal modes on a string, compute the fundamental harmonic, wave speed and tension.

Wave Animations

• *Animation: Transverse and Longitudinal Waves
• *Animation: Waves on the surface of water (a combination of transverse and longitudinal)
• *Animation:  Waves on the surface of water (peakedness depends on amplitude of circular motion)
• *Animation:  Rayleigh waves in a solid (seismic waves on a surface).
• *Animation: Wave reflections:  fixed end vs. free end, fast to slow transitions vs. slow to fast transitions.
• *Animation: Constructive and destructive interference, and formation of standing waves on a string.
• *Animation: Standing waves on a guitar string and superposition of normal modes.

*Note: this material may be taught in first or second semester physics, depending on the school.

Waves and Waves on a String

• The wave function for a wave traveling on a string [2023.01.20]
• Derivation of the 1D wave equation using partial derivatives of the wave function [2023.01.21]
• Derivation of the wave equation using Newton's 2nd law + formula for speed [2023.01.22]
• Wave function for a standing wave, boundary conditions and normal modes on a string of length L [2023.01.24]
• Energy transmitted by a wave on a string [2025.01.18]
• Potential and kinetic energy in a wave [TBA]

Introduction to Sound and Resonance with Sound

• Quick review of elastic moduli:  Young's Modulus and Bulk Modulus [2025.01.19]
• Longitudinal waves: essential concepts for sound waves [2025.01.19]
• Deriving the pressure wave function, relation between pressure and displacement amplitudes [2025.01.20]
• The speed of sound in bulk materials, thin rods and gases [TBA]
• Standing waves and normal modes for sound in an air column [TBA]

Animations

• *Animation: Math lesson: how to make a wave move to the left and right. [2021.10.02]
• *Animation: Motion perpendicular to the direction of wave velocity:  time derivatives of the wave function [2023.01.21]
• *Animation: Wave reflections:  fixed end vs. free end, fast to slow transitions vs. slow to fast transitions. (review)
• *Animation: Constructive and destructive interference, and formation of standing waves on a string. (review)

*Note: this material may be taught in first or second semester physics, depending on the school.

Energy and Intensity for Sound Waves:

• *Energy and intensity for sound waves, and the decibel (dB) scale [2025.01.25]
• *The Decibel scale and related calculations [2025.01.26]

Doppler Effect, Interference and Shock Waves:

• *The Doppler effect, derivation of equations + examples [2023.01.31]
• *Spatial interference of sound waves:  in class demonstration.
• *Time interference of sound waves:  Acoustic beats and beat frequency [2025.01.24]
• *Shock waves, the sonic boom and the angle of a shock wave + example [2025.02.08]

Animations:

• *Animation:  plane traveling at Mach 2.0 [2025.02.08]
• *Animation:  plane traveling at Mach 1.2 [2025.02.07]
• *Animation:  plane traveling at Mach 1.0 [2025.02.07]
• *Animation:  plane traveling at Mach 0.8 [2025.02.07]
• *Animation:  Beats illustration: overlapping waves arriving at the ear + composite wave arriving at the ear.  [2025.01.25]
• *Animation:  Doppler effect for a stationary source and moving listener [2023.01.29]
• *Animation:  Doppler effect for a moving source and stationary listener [2023.01.29]
• *Animation:  Sound wave interference:  acoustic beats. [2021.10.19]
• *Animation:  The sound barrier, shock waves and sonic booms [2023.02.04]
• *Animation:  spatial interference of light waves (sound is totally analogous if two speakers emit a tone!) [2021.08.03]