Combine Springs Like Capacitors, Then Use T = 2π√(m/k)
A mass on a spring oscillates with ω = sqrt(k/m). When several springs are used, first reduce them to one effective force constant, then use the same formula.
Spring-Mass Systems and Spring Combinations is the central idea in this question pattern. Identify the condition in the stem before applying the method below.
Series and parallel
A combination of effective constant K/2 and another of effective constant 2K/3 give frequencies in the ratio sqrt((K/2) / (2K/3)) = sqrt(3)/2, since f is proportional to sqrt(k).
Comparing two oscillators
For equal masses on springs of constants k1 and k2 with equal maximum speeds, v_max = A sqrt(k/m) gives A1 sqrt(k1) = A2 sqrt(k2), so the ratio of the amplitudes is A1 / A2 = sqrt(k2 / k1): the stiffer spring has the smaller amplitude.
OlympIQ exam lens
- ✓Parallel springs: k = k1 + k2; series: k = k1 k2 / (k1 + k2)
- ✓f = (1/2π) sqrt(k / m); T = 2π sqrt(m / k)
- ✓Equal maximum speeds and equal masses: amplitude is proportional to 1/sqrt(k)
Do not add spring constants for springs in series. Series springs share the same force but stretch by different amounts, so the effective constant is smaller than either spring, exactly as the effective capacitance of capacitors in series is smaller than either capacitor.
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