動力学：直前対策パック

Motion with constant acceleration: v = v₀ + a·t, s = v₀·t + ½·a·t² and v² = v₀² + 2·a·s. Pick the equation that contains the three things you know and the one you want, and keep the signs consistent with one positive direction.

Motion with constant acceleration: v = v₀ + a·t, s = v₀·t + ½·a·t² and v² = v₀² + 2·a·s. Pick the equation that contains the three things you know and the one you want, and keep the signs consistent with one positive direction.

Newton’s second law: the net force on a body equals its mass times its acceleration, ΣF = m·a. Draw a free-body diagram, choose a positive direction along the motion, and add every force in that direction: applied forces, friction μ·N against the motion, and the components of gravity.

Newton’s second law: the net force on a body equals its mass times its acceleration, ΣF = m·a. Draw a free-body diagram, choose a positive direction along the motion, and add every force in that direction: applied forces, friction μ·N against the motion, and the components of gravity.

放物運動：空気抵抗がなければ x = v_xt、y = h + v_yt − ½gt²。2 つの方向は独立で、共通なのは時間だけです。

Projectile motion: with no air resistance, x = v_xt and y = h + v_yt − ½gt². The two directions are independent except that they share the same time.

Uniform circular motion: an object moving at speed v on a circle of radius r accelerates towards the centre with a = v²/r, so it needs a net inward force F = m·v²/r. That force is supplied by something real: friction, tension, gravity or a normal force.

Uniform circular motion: an object moving at speed v on a circle of radius r accelerates towards the centre with a = v²/r, so it needs a net inward force F = m·v²/r. That force is supplied by something real: friction, tension, gravity or a normal force.

仕事とエネルギーの原理：運動エネルギーの変化は、すべての力がした仕事に等しくなります。重力とばねを位置エネルギー（mgh と ½kx²）として書くと、収支の式になります：初めのエネルギー = 終わりのエネルギー + 摩擦で失われたエネルギー (μN·d)。 動力は力 × 速さです：P = F·v。

仕事とエネルギーの原理：運動エネルギーの変化は、すべての力がした仕事に等しくなります。重力とばねを位置エネルギー（mgh と ½kx²）として書くと、収支の式になります：初めのエネルギー = 終わりのエネルギー + 摩擦で失われたエネルギー (μN·d)。

Conservation of momentum: when no external force acts during a collision, the total momentum Σm·v before equals the total after. Velocities have a direction, so pick a positive direction and give velocities the other way a minus sign. In an elastic collision kinetic energy is conserved too; in a perfectly inelastic one the bodies stick together.

Conservation of momentum: when no external force acts during a collision, the total momentum Σm·v before equals the total after. Velocities have a direction, so pick a positive direction and give velocities the other way a minus sign. In an elastic collision kinetic energy is conserved too; in a perfectly inelastic one the bodies stick together.

Rotation about a fixed axis: the sum of the moments about the axis equals I·α (ΣM = I·α), just as ΣF = m·a for translation. In energy form, a torque T turning through an angle θ (in radians) does work T·θ, which changes the kinetic energy ½·I·ω²; gravity adds or removes m·g·Δh of the centre of mass. Power is P = T·ω.

Rolling and angular momentum: a body rolling without slipping has v = ω·r, so its kinetic energy is ½·m·v² + ½·I·ω² = ½·m·v²·(1 + I/(m·r²)). Down a slope that gives a = g·sin θ/(1 + I/(m·r²)). When no external moment acts about an axis, the angular momentum about that axis stays constant, I₁·ω₁ = I₂·ω₂, even if kinetic energy is lost.

Gear geometry: the module m sets the tooth size. A gear with N teeth has a pitch diameter d = m·N, two meshing gears have the same module, and their centre distance is a = (d₁ + d₂)/2 = m·(N₁ + N₂)/2. The pitch-line speed is v = π·d·n/60, and the tangential tooth force follows from the torque: F_t = 2T/d.

歯車列：かみ合う歯車は歯の部分で同じ速さで動くので、n₁·N₁ = n₂·N₂ となり、歯数の少ない歯車ほど速く回ります。同じ軸に取り付けた歯車は同じ回転速度で回ります。動力は（損失を除いて）そのまま伝わるので、回転速度が下がるとトルクは上がります：T₂ = T₁·(N₂/N₁)·η。また P = T·ω、ただし ω = 2π·n/60。

Simple harmonic motion: a mass m on a spring of stiffness k oscillates with angular frequency ω = √(k/m) and period T = 2π/ω = 2π·√(m/k). A simple pendulum of length L has T = 2π·√(L/g). With amplitude A, the largest speed is A·ω, the largest acceleration A·ω² and the energy ½·k·A².

単振動：ばね定数 k のばねにつけた質量 m の物体は、角振動数 ω = √(k/m)、周期 T = 2π/ω = 2π·√(m/k) で振動します。長さ L の単振り子の周期は T = 2π·√(L/g) です。振幅を A とすると、最大の速さは A·ω、最大の加速度は A·ω²、エネルギーは ½·k·A² です。

質量–ばね系の自由振動：減衰のない固有角振動数は ω_n = √(k/m)（rad/s）、固有振動数は f_n = ω_n/(2π)（Hz）です。粘性ダンパ c による減衰比は ζ = c/c_c で、臨界減衰係数は c_c = 2·√(k·m) = 2·m·ω_n です。不足減衰系（ζ < 1）は ω_d = ω_n·√(1 − ζ²) で振動します。並列ばねは k = k₁ + k₂ と足し合わせ、直列ばねは逆数で 1/k = 1/k₁ + 1/k₂ と足し合わせます。

質量–ばね系の自由振動：減衰のない固有角振動数は ω_n = √(k/m)（rad/s）、固有振動数は f_n = ω_n/(2π)（Hz）です。粘性ダンパ c による減衰比は ζ = c/c_c で、臨界減衰係数は c_c = 2·√(k·m) = 2·m·ω_n です。不足減衰系（ζ < 1）は ω_d = ω_n·√(1 − ζ²) で振動します。並列ばねは k = k₁ + k₂ と足し合わせ、直列ばねは逆数で 1/k = 1/k₁ + 1/k₂ と足し合わせます。