Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (ω is any positive constant): (a) sin ωt – cos ωt (b) sin³ ωt (c) 3 cos (π/4 – 2ωt) (d) cos ωt + cos 3ωt + cos 5ωt (e) exp(–ω²t²) (f) 1 + ωt + ω²t²
((a)) Simple harmonic motion. Period = 2π/ω
((b)) Periodic but not simple harmonic. Period = 2π/ω
((c)) Simple harmonic motion. Period = π/ω
((d)) Periodic but not simple harmonic. Period = 2π/ω
((e)) Non-periodic motion
((f)) Non-periodic motion
Explanation
The textbook provides clear guidance through Example 13.3 for part (a) and Example 13.2 for identifying periodic vs non-periodic functions. For SHM, the function must be expressible as a single sine or cosine function of the form D sin(ωt + φ). Functions that are superpositions of multiple sine/cosine terms with different frequencies are periodic but not SHM. Exponential decay functions like e^(-ωt) and polynomial functions are non-periodic as they never repeat their values.
Solution Steps
Step 1: For (a) sin ωt – cos ωt, use identity: = sin(ωt – π/4). This is single sine function → SHM with T = 2π/ω
Step 2: For (b) sin³ ωt, use identity: sin³ωt = (3sin ωt – sin 3ωt)/4. Superposition of two frequencies → Periodic but not SHM. Period = 2π/ω
Step 3: For (c) 3 cos(π/4 – 2ωt) = 3 cos(2ωt – π/4). Single cosine function with angular frequency 2ω → SHM with T = 2π/2ω = π/ω
Step 4: For (d) cos ωt + cos 3ωt + cos 5ωt, superposition of three cosine functions with different frequencies → Periodic but not SHM. Period = 2π/ω (LCM of individual periods)
Step 5: For (e) exp(–ω²t²), Gaussian function decreases monotonically, never repeats → Non-periodic
Step 6: For (f) 1 + ωt + ω²t², polynomial function increases without bound, never repeats → Non-periodic