University Physics I: Mechanics

Units

1. 1

   Kinematics

   6 lessons

   1. Scalars and vectorsSplit a vector into components, add vectors component by component, and use unit vectors, the dot product and the cross product.
   2. Displacement, velocity and acceleration with derivativesFind velocity and acceleration from a position function by differentiating, and tell instantaneous values from averages.
   3. Variable acceleration with integralsFind velocity and position from a changing acceleration by integrating with initial conditions, and handle acceleration that depends on position.
   4. Representing motion: graphs, slopes and areasRead velocity and acceleration from slopes, and displacement and velocity change from signed areas, and move between x–t, v–t and a–t graphs.
   5. Reference frames and relative motionAdd velocities between reference frames as vectors, solve river and wind problems, and explain why acceleration is the same in every inertial frame.
   6. Motion in two dimensions: projectilesTreat projectile motion as independent horizontal and vertical motions joined by time, and write position, velocity and acceleration as vectors.
2. 2

   Force and translational dynamics

   13 lessons

   1. Systems and center of mass: discrete massesChoose a system, find the center of mass of discrete masses as a weighted average, and use the fact that internal forces cannot move it.
   2. Center of mass by integrationFind the mass and center of mass of a continuous body by integrating mass elements dm = λ dx, and for a plate by slicing into strips.
   3. Forces and free-body diagramsName every force on an object with its source, draw a free-body diagram of single arrows, and turn it into component equations.
   4. Newton's third law and tensionIdentify third-law pairs, draw one diagram per object, and find contact forces and the tension in a string, a pulley and a rope with mass.
   5. Newton's first law and translational equilibriumState the first law, treat equilibrium as ΣF = 0 on each axis, and solve for tensions, normal forces and friction at rest or at constant velocity.
   6. Newton's second law and multi-body systemsApply ΣF = ma to one object, to connected objects with a shared acceleration, and to a force that changes with time.
   7. Gravitational force, weight and the shell theoremUse the law of universal gravitation and the gravitational field, explain apparent weight, and apply the shell theorem inside and outside a uniform sphere.
   8. Kinetic and static frictionChoose between f_s ≤ μ_s N and f_k = μ_k N by checking whether the surfaces slip, find N from the perpendicular balance on a level floor, an incline and an angled pull, and use f_s = ma and tan θ = μ_s.
   9. Spring forces and combinations of springsUse F = −kx with x measured from the relaxed length, read k and the work ½kx² from an F against x graph, find the equivalent constant of springs in series (1/k_eq = Σ 1/kᵢ) and in parallel (k_eq = Σ kᵢ), and tell which is which.
   10. Resistive forces: setting up the equation of motionModel a resistive force as F = −kv, write m dv/dt = mg − kv for a falling body (and m dv/dt = −kv for a coasting one) with the signs checked by limits, and find the terminal velocity v_T = mg/k and the shape of v(t) from the equation.
   11. Resistive forces: solving by separation of variablesSolve m dv/dt = −kv and m dv/dt = mg − kv by separating variables, get v(t), x(t), τ = m/k and v_T = mg/k, read the 63% and 95% marks, and check any solution with four limits.
   12. Circular motion: centripetal acceleration, banked curves and orbitsUse a_c = v²/r = ω²r = 4π²r/T² toward the center, treat the centripetal force as the net inward component of real forces, and solve for a flat curve, a frictionless banked curve (tan θ = v²/(rg)), a conical pendulum and a circular orbit (v = √(GM/r), T² = 4π²r³/(GM)).
   13. Non-uniform circular motion and vertical circlesSplit the acceleration on a circle into a_t = dv/dt and a_c = v²/r, combine them as a = √(a_c² + a_t²), write the inward equation at the top (N + mg = mv²/r) and bottom (N − mg = mv²/r) of a vertical circle, find the least speed v = √(gr) at the top, and solve a conical pendulum.
3. 3

   Work, energy and power

   6 lessons

   1. Translational kinetic energyUse K = ½mv² as a scalar that grows as v², derive it by integrating the second law (∫F dx = ∫mv dv), say why it depends on the observer's frame, and tell kinetic energy from momentum with K = p²/(2m).
   2. Work by a variable forceCompute work as W = ∫F·dr, the area under the parallel component of the force, for forces that change and for constant forces at an angle, apply W_net = ΔK, and tell conservative from nonconservative work by path.
   3. Potential energy and F = -dU/dxDefine potential energy through ΔU = −∫F dx for a conservative force, find the force as F = −dU/dx (minus the slope of U), recover U from F by integrating with a chosen reference, and say why the zero of U is a free choice.
   4. Potential energy curves and equilibriumRead a U(x) graph for its equilibrium points and their stability, use K = E − U to find the allowed regions, turning points and speeds, and treat small oscillations about a minimum as a spring with k_eff = U″(x₀).
   5. Conservation of energy with and without dissipationChoose a system and write K_i + U_i + W_ext = K_f + U_f + E_thermal with E_thermal = fd, use K + U = constant and v = √(2gΔh) when only conservative forces act, and account for friction as thermal energy rather than lost energy.
   6. Power and P = F·vDefine power as the rate of energy transfer, P = dW/dt in watts, use P_avg = W/Δt and P = F·v = Fv cos θ, solve a constant-power motion from m v dv/dt = P, and find energy as the area under P against t.
4. 4

   Linear momentum

   5 lessons

   1. Linear momentum of particles and systemsUse p = mv as a signed vector, add momenta of a system to get P = Mv_cm, write Newton's second law as F_net = dp/dt, say what collision and explosion models assume, and tell momentum from kinetic energy.
   2. Impulse as the integral of forceFind the impulse of a varying force as J = ∫F dt, use J = Δp to get a velocity or an average force, read impulse and force from F-t and p-t graphs, and use F = v dm/dt for a conveyor belt.
   3. Conservation of linear momentumChoose a system, check that the net external impulse is zero, and use P_before = P_after with signs to find velocities after a push-apart or a collision, including a two-stage ballistic pendulum.
   4. Elastic and inelastic collisions in one dimensionSolve one-dimensional collisions with P always conserved, find the final velocities of an elastic collision from P and the reversal of relative velocity, find the common velocity of a perfectly inelastic one, and find the kinetic energy lost.
   5. Collisions in two dimensionsConserve momentum separately along x and y to find the velocities after a two-dimensional collision, find the common velocity when two objects lock together, and use K and the 90° rule to decide whether a collision is elastic.
5. 5

   Torque and rotational dynamics

   7 lessons

   1. Rotational kinematics with calculusDescribe rotation with θ, ω = dθ/dt and α = dω/dt in radians, find ω and θ by integrating α(t) with initial conditions, use the constant-α equations, and read slopes and areas on ω-t graphs with signs.
   2. Connecting linear and rotational motionLink the motion of a point at radius r to the rotation of its rigid body with s = rθ, v = ωr, a_t = αr and a_c = ω²r, keep the tangential and centripetal accelerations apart, and apply them to a string on a pulley.
   3. Torque as a cross productFind the torque of a force about a pivot as τ = r × F, with magnitude rF sin θ, as a signed vector in components, and add torques about one pivot to get a net torque with a clockwise or counterclockwise sense.
   4. Rotational inertia by integrationCompute rotational inertia as I = Σmr² for particles and I = ∫r² dm for rods and disks, state the axis each time, and use the results for rods, rings and disks and a rod with changing density.
   5. The parallel-axis theoremUse I = I_cm + Md² to find the rotational inertia about any axis parallel to one through the center of mass, explain why the minimum I is at the center of mass, and apply the theorem to composite bodies.
   6. Rotational equilibrium and static equilibriumState that static equilibrium needs both ΣF = 0 and Στ = 0, choose a pivot that removes unknown forces, solve for support forces on a beam, and find the friction a ladder needs.
   7. Newton's second law for rotationApply Στ = Iα about one axis, combine it with ΣF = ma through the no-slip constraint a = αR for a massive pulley, and integrate α = τ(t)/I when the torque changes with time.
6. 6

   Energy and momentum of rotating systems

   6 lessons

   1. Rotational kinetic energyUse K = ½Iω² as the sum of ½mv² over a rotating body, add ½Mv_cm² when the center also moves, and include rotation in energy conservation for a swinging rod and a falling mass on a pulley.
   2. Torque, work and rotational powerFind the work of a torque as W = ∫τ dθ in radians, equate net work to the change in ½Iω², and find rotational power as P = τω.
   3. Angular momentum and angular impulseFind the angular momentum of a particle about a point, L = mvd, and of a rigid body, L = Iω, and use Στ = dL/dt and angular impulse ∫τ dt = ΔL to find how a torque changes it.
   4. Conservation of angular momentumUse L_i = L_f when the net external torque is zero, to find angular speeds when a system changes shape or sticks together, and say what happens to kinetic energy.
   5. Rolling with and without slippingUse v_cm = ωR and K = ½Mv_cm² + ½I_cmω² for rolling without slipping, find speeds and accelerations down an incline, and explain what changes when an object slips.
   6. Orbits, energy and escape speedUse U = −GMm/r with energy and angular momentum conservation to find the speed and energies of a circular orbit, the speeds in an elliptical orbit, and the escape speed.
7. 7

   Oscillations

   5 lessons

   1. Defining simple harmonic motionDefine simple harmonic motion by the restoring force F = −kx, write the equation d²x/dt² = −ω²x with ω² = k/m, test whether a force law gives SHM, and measure x from equilibrium.
   2. Frequency and period of simple harmonic motionRelate T, f and ω, find the period of a spring-mass system and a simple pendulum, say why the amplitude does not change the period, and linearize T² against m or L.
   3. Representing and analyzing SHM: x(t), v(t), a(t)Write x = A cos(ωt + φ), differentiate it to get v and a, find the maximum speed and acceleration, sketch the three graphs from each other, and find A and φ from initial conditions.
   4. Energy of simple harmonic oscillatorsShow that an undamped oscillator keeps a constant total energy E = ½kA², find K, U and the speed at any position, write K and U as functions of time, and say how E changes with the amplitude.
   5. Simple, physical and torsion pendulumsFind the period of a physical pendulum from T = 2π√(I/(Mgd)) at small angles, recover the simple pendulum as a special case, and find the period of a torsion pendulum from T = 2π√(I/κ).
8. 8

   Gravitation and orbits

   4 lessons

   1. Universal gravitation and the shell theoremWrite Newton's law of gravitation as a vector, add the pulls of several masses with the field g = GM/r², and use the shell theorem to find g outside, inside a shell and inside a solid sphere.
   2. Gravitational potential energy and escape speedDerive U = −GMm/r from the work integral with its zero at infinity, show that mgh is its near-surface form, read bound and unbound motion from the sign of E, and find the escape speed √(2GM/R).
   3. Kepler's laws and orbital energyState Kepler's three laws, derive the equal-areas law from conserved angular momentum and T² = 4π²a³/GM for circular orbits, and use E = −GMm/2a to find speeds, energies and the burns of a transfer orbit.
   4. Central-force motion and the effective potentialUse conserved angular momentum and energy to reduce motion under any central force to one radial equation, read turning points and circular orbits from the effective potential, and tell the conic orbits of an inverse-square force from the sign of E.