Problem Statement
Solve the oscillation/wave problem: Solve the oscillation/wave problem: Write the equation of a plane harmonic wave traveling in the $+x$ direction. Define all parameters. The general plane harmonic wave: $$\boxed{\xi(x,t) = a\cos\left(\omega t – kx + \phi_0\right) = a\cos\left(\frac{2\pi t}{T} – \frac{2\pi x}{\lambda} + \phi_0\right)
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
Conservation of linear momentum holds whenever the net external force on a system is zero. In collisions, momentum is always conserved. Additionally, in elastic collisions kinetic energy is also conserved, whereas in perfectly inelastic collisions the objects stick together and kinetic energy is partially converted to heat and deformation.
- $\mathbf{p}_\text{tot} = \sum m_i\mathbf{v}_i = \text{const}$ — conservation of momentum
- Elastic: $\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 = \text{const}$ — KE conserved
- Inelastic: $m_1v_1 = (m_1+m_2)V$ — perfectly inelastic
- $\eta = \Delta KE/KE_0 = M/(m+M)$ — fractional KE loss (bullet-block)
Step-by-Step Solution
Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$\boxed{\xi(x,t) = a\cos\left(\omega t – kx + \phi_0\right) = a\cos\left(\frac{2\pi t}{T} – \frac{2\pi x}{\lambda} + \phi_0\right)
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:
$$
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Answer
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Answer
$$\boxed{T = 2\pi\sqrt{m/k}}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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