https://teddyrookbook.com/physics/mechanics/Recent content in Mechanics with Calculus on Teddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenhttps://teddyrookbook.com/physics/mechanics/velocity-derivative/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/velocity-derivative/v = dx/dt gives the velocity at any instant from the position function. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/acceleration-derivative/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/acceleration-derivative/a = dv/dt = d²x/dt², and zero velocity at an instant says nothing about the acceleration. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/motion-from-acceleration/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/motion-from-acceleration/Integrate a(t) once for v(t) and again for x(t), using the starting values to fix the constants. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/derive-kinematics/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/derive-kinematics/Integrating a constant acceleration twice gives the familiar equations and shows exactly when they apply. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/acceleration-of-position/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/acceleration-of-position/When a depends on x, write a = v dv/dx and integrate over x. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/motion-vectors/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/motion-vectors/r(t), v = dr/dt and a = dv/dt describe motion in two and three dimensions, one component at a time. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/projectile-optimization/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/projectile-optimization/Use derivatives to find the highest point, the range and the launch angle that goes farthest. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/galilean-frames/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/galilean-frames/Frames moving at constant velocity disagree about velocities but agree about accelerations and forces. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/tangential-normal-acceleration/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/tangential-normal-acceleration/On a curve, acceleration splits into dv/dt along the path and v²/r toward the center. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/time-varying-force/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/time-varying-force/Integrate F(t)/m to find the velocity when the force changes with time. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/linear-drag/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/linear-drag/With drag F = −bv, the speed approaches v_T = mg/b as v(t) = v_T(1 − e^(−t/τ)), with τ = m/b. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/quadratic-drag/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/quadratic-drag/Fast objects in air feel drag ½CρAv², which gives a terminal speed √(2mg/(CρA)). Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/banked-curves/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/banked-curves/Find the design speed of a banked road and the range of speeds friction allows. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/pulley-constraints/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/pulley-constraints/The fixed length of a rope links the accelerations, so a movable pulley gives a₁ = 2a₂. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/center-of-mass-integral/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/center-of-mass-integral/x_cm = (1/M)∫x dm, with dm = λ dx for a rod whose density varies. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/work-integral/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/work-integral/W = ∫F dx, the area under the force–position curve. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/work-along-path/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/work-along-path/W = ∫F · dr adds up only the part of the force along each small step of the path. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/work-energy-derivation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/work-energy-derivation/Integrating m dv/dt over x turns F = ma into W_net = ΔK. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/conservative-forces/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/conservative-forces/A force is conservative when its work doesn't depend on the path, so its work around any closed loop is zero. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/potential-from-force/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/potential-from-force/U(x) = −∫F dx, which gives ½kx² for a spring and mgy for uniform gravity. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/force-from-potential/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/force-from-potential/F = −dU/dx: the force points downhill on the potential energy curve. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/energy-diagrams/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/energy-diagrams/Read turning points, allowed regions and stable or unstable equilibria from a graph of U(x). Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/instantaneous-power/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/instantaneous-power/P = dW/dt = F · v gives the power at each instant. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/escape-speed/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/escape-speed/From U = −GMm/r, the escape speed is √(2GM/r), the same for any mass. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/impulse-integral/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/impulse-integral/J = ∫F dt for any force that changes in time. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/elastic-collisions-1d/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/elastic-collisions-1d/Combine conservation of momentum and of kinetic energy: the relative speed reverses in a one-dimensional elastic collision. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/collisions-2d/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/collisions-2d/Conserve the x- and y-components of momentum separately. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/system-of-particles/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/system-of-particles/ΣF_external = M·a_cm, because the internal forces cancel in pairs. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/rocket-equation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/rocket-equation/A rocket gains Δv = v_e ln(m₀/m_f) by throwing mass backward. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/angular-calculus/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/angular-calculus/ω = dθ/dt and α = dω/dt; integrate α(t) to find ω and θ. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/moment-of-inertia-integral/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/moment-of-inertia-integral/I = ∫r² dm gives the rotational inertia of rods, rings, disks and spheres. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/parallel-axis/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/parallel-axis/I = I_cm + Md² for an axis shifted a distance d from the center of mass. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/torque-cross-product/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/torque-cross-product/τ = r × F, with its direction given by the right-hand rule. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/rolling-dynamics/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/rolling-dynamics/Static friction makes rolling work: on a ramp a = g sin θ/(1 + I/(MR²)). Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/skidding-to-rolling/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/skidding-to-rolling/A ball that starts out sliding skids until v = Rω, losing energy to kinetic friction on the way. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/angular-momentum-particle/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/angular-momentum-particle/L = r × p, so even a particle moving in a straight line has angular momentum about a point. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/rotational-collisions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/rotational-collisions/When a ball sticks to a pivoted rod, angular momentum about the pivot is conserved but linear momentum is not. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/rotational-work-power/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/rotational-work-power/W = ∫τ dθ and P = τω. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/precession/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/precession/A torque at right angles to the spin turns the direction of L, so a top precesses at Ω = τ/L. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/statics-problems/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/statics-problems/For ladders, hinged beams and hanging signs, choose a pivot that removes the unknowns you don't need. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/shell-theorem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/shell-theorem/Outside a uniform sphere, gravity acts as if all the mass sat at the center; inside, g grows in proportion to r. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/orbital-energy/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/orbital-energy/A circular orbit has E = −GMm/(2r), so climbing to a higher orbit takes energy yet leaves you moving more slowly. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/kepler-second-from-newton/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/kepler-second-from-newton/Gravity exerts no torque about the Sun, so L is constant and equal areas are swept in equal times. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/kepler-third-from-newton/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/kepler-third-from-newton/For a circular orbit T² = 4π²r³/(GM), which weighs a planet or star from its satellites. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/vis-viva/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/vis-viva/v² = GM(2/r − 1/a) gives the speed anywhere on an ellipse, which plans transfer orbits. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/shm-equation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/shm-equation/x″ = −ω²x is solved by x = A cos(ωt + φ) with ω = √(k/m). Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/shm-initial-conditions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/shm-initial-conditions/Use x(0) and v(0) to find the amplitude A and the phase φ. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/vertical-spring/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/vertical-spring/Gravity only moves the equilibrium point; the period stays 2π√(m/k). Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/physical-pendulum/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/physical-pendulum/A swinging rigid body has T = 2π√(I/(mgd)), and small angles make any pendulum simple harmonic. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/damped-oscillations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/damped-oscillations/Drag makes the amplitude shrink as e^(−bt/2m), and too much damping stops the swinging altogether. Mechanics with Calculus · Physics Ladderhttps://teddyrookbook.com/physics/mechanics/driven-resonance/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/mechanics/driven-resonance/Pushing near the natural frequency builds a large amplitude, limited only by damping. Mechanics with Calculus · Physics Ladder