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The spring-mass system is the purest realization of SHM: $T = 2\pi\sqrt{m/k}$, independent of gravity, amplitude, and surface orientation. A vertical spring has the same time period as a horizontal one — gravity merely shifts the equilibrium by $mg/k$ without altering the oscillation dynamics. For spring combinations: series gives $k_{\text{eff}} = k_1k_2/(k_1+k_2)$ (softer, longer $T$) and parallel gives $k_{\text{eff}} = k_1 + k_2$ (stiffer, shorter $T$).

Spring cutting is a critical concept: cutting a spring of constant $k$ and length $L$ into pieces yields constants inversely proportional to piece lengths. A piece of length $L/n$ has constant $nk$. For a cut in ratio $m:n$, the constants are $k(m+n)/m$ and $k(m+n)/n$. When springs are cut and reconnected, first find individual constants, then apply series/parallel rules. A useful shortcut for vertical springs: if the static extension at equilibrium is $d$ (where $mg = kd$), then $T = 2\pi\sqrt{d/g}$ — eliminating the need to know $m$ or $k$ separately. For two masses $m_1, m_2$ on a spring, use reduced mass $\mu = m_1m_2/(m_1+m_2)$.