A 0.30 kg mass is suspended on a spring. In equilibrium the mass stretches the spring 2.0 cm downward. The mass is then pulled an additional distance of 1.0 cm down and released from rest. Calculate the period of oscillation.

Question

BlackTom AI · Accepted Answer

Hooke’s law and the period formula for a mass-spring simple harmonic oscillator give the solution: T = 2π√(m/k).

1) Relevant concepts
- Static equilibrium stretch x_eq satisfies mg = k x_eq (Hooke’s law balanced by weight), so the spring constant k can be found from k = mg/x_eq.
- The period of small oscillations for a mass m on a spring (linear, ideal spring) is T = 2π√(m/k). The period is independent of amplitude for linear SHM.

2) Step-by-step reasoning and calculations
- Given: m = 0.30 kg, equilibrium stretch x_eq = 2.0 cm = 0.020 m, g = 9.80 m/s^2.
- Find k from mg = k x_eq:
  mg = 0.30 kg × 9.80 m/s^2 = 2.94 N.
  k = mg / x_eq = 2.94 N / 0.020 m = 147 N/m.
- Compute the period T:
  T = 2π √(m/k) = 2π √(0.30 / 147).
  Intermediate: 0.30 / 147 = 0.0020408163...
  √(0.0020408163) = 0.045177... s.
  T = 2π × 0.045177... = 0.2838... s ≈ 0.28 s.

3) Why this is correct
- The equilibrium stretch gives the spring constant directly via mg = kx. Once k is known, the standard SHM period formula yields T. The additional 1.0 cm displacement is just the amplitude of oscillation and does not affect the period for a linear spring, so the period depends only on m and k.

4) Why the other options are incorrect
- 0.14 s is about half the correct value (would correspond to using an incorrect k or mis-evaluating the square root by a factor of 2).
- 0.020 s and 0.078 s are far too small; they correspond to orders-of-magnitude errors in either k or the square-root step or using incorrect units (e.g., forgetting conversion from cm to m).

Conclusion: The calculated period is approximately 0.28 s, which matches the provided correct answer (0.28 s).

A 0.30 kg mass is suspended on a spring. In equilibrium the mass stretches the spring 2.0 cm downward. The mass is then pulled an additional distance of 1.0 cm down and released from rest. Calculate the period of oscillation.单项选择题

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