Problem Statement
Solve the oscillation/wave problem: In Kundt’s tube experiment, cork dust forms nodes at every 15 cm in air and every 7.0 cm in an unknown gas. Ratio of wave speeds and gas identification? In Kundt’s tube, standing waves form and cork dust collects at nodes. The node spacing = $\lambda/2$. $$\lambda_{\rm air}/2 = 15 \text{ cm} \implie
Given Information
- Mass $m$ and spring constant $k$ (or equivalent), or wave parameters
- Initial conditions (amplitude $A$, phase $\phi$) as given
Physical Concepts & Formulas
Simple harmonic motion arises whenever a restoring force is proportional to displacement: $F = -kx$. Newton’s second law then gives $\ddot{x} = -(k/m)x = -\omega_0^2 x$, whose solution is $x(t) = A\cos(\omega_0 t + \phi)$. The total mechanical energy $E = \frac{1}{2}kA^2$ is constant for ideal SHM. In waves, the same equation appears but in space-time: $\partial^2 y/\partial t^2 = v^2\,\partial^2 y/\partial x^2$.
- $\omega_0 = \sqrt{k/m}$ — angular frequency
- $T = 2\pi/\omega_0 = 2\pi\sqrt{m/k}$ — period
- $x(t) = A\cos(\omega_0 t + \phi)$ — general SHM solution
- $E = \tfrac{1}{2}kA^2$ — total mechanical energy
- $v = f\lambda$ — wave speed
Step-by-Step Solution
Step 1 — Identify the restoring force and write the equation of motion.
Step 2 — Find $\omega_0$: $\omega_0 = \sqrt{k/m}$
Step 3 — Apply initial conditions to find $A$ and $\phi$.
Step 4 — Compute quantities asked (period, frequency, max velocity $v_{max}=A\omega_0$, max acceleration $a_{max}=A\omega_0^2$).
Worked Calculation
Substituting all values with units:
$$T = 2\pi\sqrt{\frac{m}{k}}\quad,\quad v_{\max} = A\omega_0 = A\sqrt{\frac{k}{m}}$$
Answer
$$\boxed{T = 2\pi\sqrt{m/k}}$$
Physical Interpretation
The period of a spring-mass oscillator depends only on $m$ and $k$ — not on the amplitude. This isochronous property is what made pendulum clocks reliable for centuries: large and small swings take the same time (for small angles).
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