Classical Mechanics

Foundations & Vectors

01SI Units and the Constants That Define Them 02Defining the Second and the Meter 03Redefining the Kilogram via Planck's Constant 04Defining SI Units from Fixed Constants 05Radians, Steradians & Small Angles 06Dimensional Analysis: M, L, and T 07Deriving Laws from Dimensions 08Fermi Estimation Problem set0/10 Problem set 20/10 Practice∞

01Vector Addition and Scalar Multiplication 02Cartesian and Cylindrical Coordinates 03Vector Components in Cartesian Coordinates 04Solving Vector Problems with Components 05Vector Components Under Coordinate Rotation 06The Cross Product 07The Cross Product Formula Problem set0/10 Problem set 20/10 Practice∞

Kinematics

01Position, Time Intervals, and Displacement 02Instantaneous Velocity and the Derivative 03Acceleration as the Second Derivative 04Kinematic Equations from Graph Areas 05Integration and the Fundamental Theorem 06Polynomial and Piecewise Acceleration Problem set0/10 Problem set 20/10 MIT problem set0/4 Practice∞

01Independent Motion Along x and y Axes 02Setting Up a Projectile Motion Problem 03Time of Flight and Maximum Height 04Eliminating Time to Find the Parabola 05Aim Straight at the Falling Bucket Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

01Centripetal Acceleration and Central Motion 02Polar Unit Vectors and the Velocity of Circular Motion 03Deriving v = rω from Chord Length and Small Angles 04Radial and Tangential Acceleration in Polar Coordinates 05Uniform Circular Motion: Period, Frequency, and v²/R 06Angular Velocity and Acceleration as Vectors 07Variable Radius: Radial Velocity and the Coriolis Term Problem set0/10 Problem set 20/10 Practice∞

Dynamics

01Defining Force: Mass, Acceleration, and Force as a Vector 02Newton's First Law: Inertia and Inertial Reference Frames 03Newton's Second and Third Laws: Force as the Rate of Change of Momentum Problem set0/10 Problem set 20/10 Practice∞

Force Laws

01Force Laws and Hooke's Law: Measuring the Spring Force 02Universal Gravitation and the Principle of Equivalence 03Electric Charge, Coulomb's Law, and Newton's Third Law Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

Constraint Forces

01The Normal Force: Why It Isn't Your Weight 02Static and Kinetic Friction and the f = μN Laws 03Free-Body Diagrams and Building a Physical Model 04Tension in a Rope: Massless vs. Massive 05Tension in a Hanging Rope: Two Methods Problem set0/10 Problem set 20/10 MIT problem set0/4 Practice∞

Applying Newton's Laws

01The Staircase Problem: Friction, Launch, and Landing 02Cart, Pulley, and Hanging Block: Finding the Tension 03Inextensible Strings: Constraints in a Pulley System 04A Block on an Accelerating Frictionless Wedge 05The Capstan Equation: Rope Friction Around a Post 06Terminal Velocity and the Hyperbolic-Tangent Law Problem set0/10 Problem set 20/10 MIT problem set0/4 Practice∞

Fluid Resistance

01The Drag Force: Two Regimes of Fluid Resistance 02Stokes' Law: Terminal Velocity of a Falling Sphere 03Quadratic Drag: The Reciprocal Velocity Decay Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

Circular Motion & Gravitation

01Newton's Second Law for Circular Motion 02Universal Gravitation: Deriving the Moon's Period 03Geostationary Satellites and Binary Star Orbits 04Tension in Circular Motion: From Masses to a Rope 05Circular Motion on a Cone and a Turntable 06The Shell Theorem: Gravity Acts From the Center Problem set0/10 Problem set 20/10 MIT problem set0/6 Practice∞

Momentum & Center of Mass

01Impulse and Momentum: Newton's Second Law as an Integral 02Drawing the System Boundary: Why Only External Forces Count 03Center of Mass: From Two Particles to a Continuous Rod 04Center-of-Mass Motion and Conservation of Momentum 05Setting Up Any Momentum Problem: System, Axes, Diagrams 06The Exploding Projectile: Where the Larger Fragment Lands 07Inelastic Collision Then Friction: The Plane and the Sandbag Problem set0/10 Problem set 20/10 MIT problem set0/5 Practice∞

Relative Motion & Variable Mass

01Reference Frames and the Galilean Transformation 02Relative Velocity Worked Examples: Circle and Cart 03Feynman, Challenger, and the Limits of a Curve Fit 04Momentum with Flowing Mass: The Four Categories 05Leaking Sand vs. Blown-In Grain: Two Freight Cars 06The Boat and the Fire Hose: Mass In, Momentum In 07Deriving the Rocket Equation: Where Thrust Comes From 08Rocket Speed, Mass Ratio, and Why Rockets Stage Problem set0/10 Problem set 20/10 MIT problem set0/5 Practice∞

Energy & Work

01Conservation of Energy: System, Boundary, and Surroundings 02Kinetic Energy: From Speed Squared to Force Times Distance 03Work Done by a Constant Force: Positive, Negative, or Zero 04Work as Area Under the Curve and the Work-Energy Theorem 05Power: Average, Instantaneous, and Force Times Velocity Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

Work in Three Dimensions

01The Scalar Product: Work as Force Dot Displacement 02Work as a Line Integral Along Any Curved Path 03Inverse-Square Forces, 3D Work-Energy, and Power 04Reduced Mass and the Work Inside a Two-Body System Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

Conservative Forces & Potential Energy

01Conservative Forces: Why Gravity's Work Ignores the Path 02The Closed-Loop Theorem: Why Conservative Forces Give Energy Back 03Defining Potential Energy from Newton's Third Law 04Three Potential Energies from One Recipe, and Where the Zero Goes Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

Conservation of Mechanical Energy

01Conservation of Mechanical Energy: Closed Systems, Conservative Forces 02The Spring Energy Diagram: Turning Points and Forbidden Regions 03Stable vs Unstable Equilibrium on a Cubic Energy Diagram 04Non-Conservative Forces: When Mechanical Energy Is Not Conserved 05Dissipation and the System Boundary: Where Friction's Energy Goes Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

Energy Methods in Practice

01Escape Velocity: Zero Total Energy on Asteroid Toro 02Loop-the-Loop: Pairing Energy with Newton's Second Law 03Friction That Grows With Distance: The Energy Integral 04Maximum Spring Compression on a Frictional Incline 05Sliding Off a Sphere: The Angle Where Contact Breaks Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

Collision Theory

01Relative Velocity, Reduced Mass, and the Center-of-Mass Frame 02Coefficient of Restitution and the One-Dimensional Elastic Collision 03Elastic Collisions Reverse in the Center-of-Mass Frame 04Stacked Superballs: Why the Small Ball Rises Nine Times Higher 05Two-Dimensional Elastic Collisions in the Laboratory Frame 06The Right-Angle Rule for Equal Masses, via the Dot Product 07From Center-of-Mass Angle to Laboratory Angle in Scattering Problem set0/10 Problem set 20/10 MIT problem set0/6 Practice∞

Fixed-Axis Rotation

01Rigid Bodies and Fixed-Axis Rotation 02Angular Signs: Speeding Up or Slowing Down 03Rotational Kinetic Energy and Moment of Inertia 04The Uniform Disc and the Parallel Axis Theorem 05Energy Conservation with a Massive Pulley 06The Rod-and-Disc Pendulum: Speed at the Bottom 07Proving the Parallel Axis Theorem Problem set0/10 Problem set 20/10 MIT problem set0/3 Practice∞

Rotational Dynamics

01Torque About a Point: Moment Arm and Sign Conventions 02Torque by Components: A Tilted Lever and the Ankle on Tiptoe 03Deriving τ = Iα: Newton's Second Law for Rotation 04Center of Gravity and Unequal Tensions on a Massive Pulley 05Measuring a Rotor's Moment of Inertia from Two Runs Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

Rotational Work & Static Equilibrium

01Torque Through an Angle: Rotational Work and Power 02Static Equilibrium and the Lever Law 03The Generalized Lever Law: Forces at an Angle 04The Hinged Rod and the Person on a Hill 05The Knee Joint and Torque About Any Point Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

Leaking Sand vs. Blown-In Grain: Two Freight Cars

Why does sand leaking out leave the equation of motion untouched while grain blown in adds an entirely new term?

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Your summary note

1. Equation of motion $F = m_c(t)\frac{dv_c}{dt}$ for leaking sand
  Form the system momentum at both instants, show the mass-change and velocity-change contributions cancelling, and substitute $m_c(t) = m_c + m_s - bt$ into the momentum principle.
2. Integrating to a logarithm, and the emptied car
  Separate variables and integrate from rest to $v_c(t) = \frac{F}{b}\ln\frac{m_c+m_s}{m_c+m_s-bt}$ , then evaluate at $t = m_s/b$ for $v_c = \frac{F}{b}\ln\frac{m_c+m_s}{m_c}$ .
3. The extra term $\frac{dm_A}{dt}(v_A - u)$ for arriving grain
  Expand the momentum difference for car $A$ with $\Delta m_A = \Delta m_g$ , drop the second-order product $\Delta m_A\Delta\vec{v}_A$ in the limit, and reach $0 = m_A\frac{dv_A}{dt} + \frac{dm_A}{dt}(v_A - u)$ .
4. The grain result and its momentum check
  Integrate by separation of variables with $m_A(t) = m_{A,0} + bt$ to get $v_A(t) = \frac{btu}{m_{A,0}+bt}$ , then rewrite it as $(m_{A,0}+bt)v_A = btu$ .
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