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Physics · Simple harmonic motion (S.H.M.) and its equation, phase

A particle moves along the X-axis according to the equation . \) the amplitudes

A particle moves along the X-axis according to the equation \( \boldsymbol{x}= \) \( 10 \sin ^{3}(\pi t) . \) the amplitudes and frequencies of component SHMs are

  • A. amplitude \( 30 / 4,10 / 4 ; \) frequencies3 \( / 2,1 / 2 \)
  • B. amplitude30/4,10/4; frequencies1/2,3/2
  • C. amplitude10, \( 10 ; \) frequencies1/2,1/2
  • D. amplitude30/4,10; frequencies3/2,

Step-by-step solution

Using the identity sin^3θ = (3 sinθ - sin3θ)/4, the given equation x = 10 sin^3(π t) becomes x = (30/4) sin(π t) - (10/4) sin(3π t). This represents two SHMs: one with amplitude 30/4 and angular frequency π (frequency = π/(2π) = 1/2), and another with amplitude 10/4 and angular frequency 3π (frequency = 3π/(2π) = 3/2). Thus the amplitudes are 30/4 and 10/4, and frequencies are 1/2 and 3/2.
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