Advanced — Energy Method for OscillationHardQuestion 30476
Question
A mass m on a spring (k) on a surface inclined at θ (rough, μ). Energy dissipated per oscillation when amplitude reduces from A to A' is:
- A½k(A²−A'²)
- B½k(A−A')²
- C2μmg cosθ(A+A')Correct
- DμmgA cosθ
Correct answer
2μmg cosθ(A+A')
Explanation
Per half-cycle, friction force μmg cosθ acts over distance (A+A')/2... over a full cycle, displacement of the block is 2(A+A') approximately, and friction does work 2μmg cosθ×(A+A')/2×2... The energy dissipated per full cycle = 4μmg cosθ × A_avg. A more careful analysis gives the dissipation per cycle = 4μmg cosθ×A for amplitude A.