Oscillations formulas

Mass bouncing on a spring (sideways or hanging). Know f? Type 1/f as T. Know ω? Type 2π/ω as T. Need ω? For a spring ω = √(k/m): type it right in any ω box, e.g. √(200/0.5). Only told how far a hanging mass stretched it? Get k from mg = kΔy first. Mass added to change T? Solve m for each T and subtract. Twisting (torsion) pendulum: type I as m and the torsion constant as k.

Ball on a string, small swings. The mass does not matter. Find g (other planet): press CLEAR on the g box (9.8) and leave it empty. Swinging rod, pivot at one end: type 2/3 of the rod length as L. Any swinging object: type I/(m*d) as L (d = pivot to center of mass). Hanging spring that stretched Δy: type Δy as L. Pendulum ω = √(g/L).

Period T = seconds for 1 full back-and-forth. Frequency f = back-and-forths per second (Hz). Counted N cycles in t seconds? Type t/N as T, e.g. 20/10. End to middle takes T/4; end to end takes T/2. Type 4t or 2t as T. Know ω? Type ω/(2π) as f to get T.

ω (rad/s) is what goes in x = A cos(ωt + φ) and the speed formulas. ω is not f. Know T? Type 1/T as f, e.g. 1/0.5. Spring: ω = √(k/m). Pendulum: ω = √(g/L). Type these right in the ω box, e.g. √(200/0.5), to get f.

Mass hanging still on a spring: Δy = how far it stretched from its natural length. Use it to find k, then the period. Pulled down more and let go? That extra pull is the amplitude A, not Δy. Spring force at a stretch x: F = kx (Work & energy). Max force = kA. No mass given? Period = 2π√(Δy/g): use T = 2π√(L/g) with Δy as L.

Position at time t. Angles here are RADIANS (π = 180°). φ: pulled to +A and let go → 0. At the middle moving + → −π/2. At −A → π. Given x = A sin(ωt)? Use φ = −π/2. Given x = 0.25 cos(0.4πt)? A = 0.25, ω = 0.4π, φ = 0. t answer = first time it is at x; the 2nd answer is the next time. Starts at x₀ moving at v₀? Get A from v = ω√(A² − x²), then t = 0, leave φ empty: pick the φ with the opposite sign to v₀.

Speed at a spot x (x measured from the middle). Max speed (at the middle): type 0 as x. That is vₘₐₓ = Aω. Amplitude from a speed at a spot: leave A empty. ω: type √(k/m) for a spring, 2π/T, or 2πf. Speed is 0 at the ends (x = A).

Acceleration at a spot x. It always points back to the middle (opposite sign to x). Max acceleration aₘₐₓ = Aω²: type A as x and drop the minus sign. Know aₘₐₓ and A? Type them as a and x, leave ω empty. At a time t: find x first with x = A cos(ωt + φ). Spring: type √(k/m) as ω.

Total energy. It stays the same the whole time. Spring energy at a spot x: type x as A. Then kinetic energy = E − that. Kinetic energy ½mv²: type m in the k box and v in the A box. Max speed from E: type m as k and leave A empty (answer is vₘₐₓ). No k? Type m*ω² as k.

Velocity at time t. + or − shows the direction. RADIANS, and the same φ as in x = A cos(ωt + φ). vₘₐₓ = Aω: type it as A*ω, e.g. 0.1*20. t answer = first time it has that velocity. ω: type 2π/T, 2πf or √(k/m).

Damped (friction slows it): the amplitude shrinks over time. Only told a fraction? Half the amplitude: type 1/1 as A₀ and 1/2 as A. Energy goes with A²: half the energy → type 1/1 as A₀ and √(1/2) as A. Critical damping (no bouncing): b = 2√(mk). Damped ω = √(k/m − (b/2m)²): type it in an ω box.