A spring-mass system oscillates in SHM with period T. The amplitude is then tripled, with mass and spring constant unchanged. What is the new period?
AT/3
BT/√3
CT
D3T
Period in SHM is T = 2π√(m/k), which depends only on mass and spring constant — not on amplitude. Tripling the amplitude means the system travels farther, but the restoring force also grows proportionally (F = −kx), so the system moves faster over the larger distance in exactly the same time. This amplitude-independence is a defining and non-obvious feature of SHM.
Question 2 True / False
A system oscillates with angular frequency ω = 2π rad/s. This means the system completes 2π full oscillations per second.
TTrue
FFalse
Answer: False
Angular frequency ω and ordinary frequency f are related by ω = 2πf. If ω = 2π rad/s, then f = ω/(2π) = 1 Hz — the system completes exactly one full oscillation per second. Angular frequency measures radians per second, and one full cycle is 2π radians. Confusing ω with f introduces an off-by-2π error that propagates into period and energy calculations.
Question 3 Short Answer
Starting from Newton's second law applied to a restoring force F = −kx, explain how you conclude that the motion must be sinusoidal.
Think about your answer, then reveal below.
Model answer: Applying F = ma gives m(d²x/dt²) = −kx, or d²x/dt² = −(k/m)x = −ω²x. The solutions to this ODE are functions whose second derivative equals −ω² times themselves — sines and cosines. The general solution x(t) = A cos(ωt + φ) with ω = √(k/m) satisfies the equation exactly.
This is a second-order linear ODE with constant coefficients. The characteristic equation r² = −ω² has purely imaginary roots ±iω, giving complex exponential solutions e^(±iωt), which in real form are cos(ωt) and sin(ωt). The linearity of the ODE guarantees that any linear combination A cos(ωt + φ) is also a solution, with constants A and φ set by initial conditions.