Problem Statement
Solve the oscillation/wave problem: A harmonic oscillator is driven by force $F = F_0\cos(\omega t)$. Find the steady-state amplitude and phase. The equation of motion: $$m\ddot{x} + r\dot{x} + kx = F_0\cos(\omega t)$$ The steady-state solution is $x = A\cos(\omega t – \psi)$ where: $$A = \frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2 + 4
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
Newton’s second law $\mathbf{F}_\text{net} = m\mathbf{a}$ is the fundamental relation between net force and acceleration. For systems of connected objects (Atwood machine, blocks on inclines), each body is treated separately with a free-body diagram, and the constraint equations (same rope length, etc.) link the accelerations.
- $\mathbf{F}_{\text{net}} = m\mathbf{a}$ — Newton’s second law
- Atwood: $a = (m_1-m_2)g/(m_1+m_2)$, $T = 2m_1m_2g/(m_1+m_2)$
- $f_k = \mu_k N$ — kinetic friction
Step-by-Step Solution
Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$m\ddot{x} + r\dot{x} + kx = F_0\cos(\omega t)$$
$$A = \frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2 + 4
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:
$$
$$
Answer
$$
Answer
$$\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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