The period of a spring's oscillation follows from its equation of motion. A comparison with a pendulum shows when the period is independent of amplitude.
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同じばねとおもりで振幅を9倍にすると、1往復の時間は？
長さ1 mの振り子を10°と90°から静かに放す。周期が短いのは？
The equations explain the phenomenon
1The spring force is substituted into the equation of motion.m d²xdt²=−k x……
2The equation is rearranged to give the acceleration, with ω²=k/m.d²xdt²=−ω² x……
3Differentiating the equation for the projection of circular motion 2 times shows whether it satisfies the equation of motion.x=A sin(ωt + φ)……
4The period is the time required for the phase to advance by 2π.T=2π√(mk)……
5The equation of motion for the pendulum is established.m L d²θdt²=−m g sinθ……
6The equation is approximated for small swing angles.Period in the small-angle approximation:T=2π√(Lg)……
7Large swings are analyzed without approximating sinθ.T(θ₀)＞2π√(Lg)……
Further topics: velocity and acceleration, vertical springs, and energy
8Differentiating the position equation gives the velocity and acceleration.v=Aω cos(ωt + φ)，a=−Aω² sin(ωt + φ)……
9The equilibrium position of the vertical spring is found, and the equation is rewritten in terms of displacement from equilibrium.With y=x − mgk,m d²ydt²=−k y……
10Multiplying the equation of motion by velocity gives conservation of energy.½ m v²+½ k x²=½ k A²……
What this chapter has shown
The restoring force of an ideal spring is proportional to displacement. The period depends on the mass and spring constant and is independent of amplitude.
1.000 times as longPeriod ratio (9A to A)
When a pendulum is released from rest at 90°, its period is 1.18 times the period for release at 10°.
In everyday lifeA pendulum clock uses small swings, so slight changes in amplitude have little effect on its timekeeping.
The equation of motion with the spring force −kx has the same form as the equation for the projection of circular motion. Its solution is therefore a trigonometric function, and the period 2π√(m/k) is independent of amplitude. A pendulum is approximately described by the same equation only for small oscillations, when its period can be taken as 2π√(L/g). For large oscillations, the approximation sinθ ≒ θ no longer holds, and the period increases.