Problem 1
At forcing frequency ω=ω₀=√(k/m), producing unbounded linear growth in the ideal model.
Faculty use
Sinusoidal Forcing and Resonance
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
It limits amplitude, broadens the peak, and shifts the frequency of maximum displacement slightly below ω₀.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind separating transient response from periodic steady state and measuring amplitude. Use the formulas A(ω)=F₀/√((k−mω²)²+(cω)²); phase lag tanφ=cω/(k−mω²).
Problem 4
A strong error analysis identifies the first step that violates classification, algebra, theorem hypotheses, units, initial/boundary data, or numerical assumptions; then repairs that step and re-verifies the final result.
Problem 5
Application responses vary. Require defined variables, a compatible differential-equation model connected to machines, buildings, vehicles, sensors, acoustic systems, and electrical circuits, a complete computation, interpretation, and limitations.
Problem 6
Accept two independent checks appropriate to the task: differentiation/substitution, initial or boundary conditions, dimensional analysis, phase-line or stability evidence, residuals, step refinement, conservation/energy checks, data comparison, or a second method.
Scoring guidance
Six problems × 10 points: 5 mathematical accuracy, 2 structure/justification, 2 verification/interpretation, 1 notation and communication.