Worked exercise — Maximum velocity from energy

Problem

Same mass-spring system as the previous exercise ($K = 50$ N/m, $A = 8{,}1$ cm, $m = 0{,}2$ kg, $\omega \approx 15{,}8$ rad/s). Calculate the total energy and the maximum velocity, checking the consistency of the energy balance.

Solution

Total energy. It depends only on the amplitude: $E = \tfrac{1}{2}K A^2 = 0{,}5\cdot 50\cdot (0{,}081)^2 \approx \ev{0{,}164 \ \text{J}}$

Maximum velocity. It is reached at the passage through equilibrium, where all the energy is kinetic: $v_\text{max} = A\omega = 0{,}081\cdot 15{,}8 \approx \ev{1{,}28 \ \text{m/s}}$

Balance check. At equilibrium $E = \tfrac{1}{2}m v_\text{max}^2$: $\tfrac{1}{2}\cdot 0{,}2\cdot (1{,}28)^2 \approx 0{,}164 \ \text{J}$ The same value found as $\tfrac{1}{2}K A^2$: the balance closes, the energy is exchanged entirely between elastic form (at the extremes) and kinetic (at equilibrium).

Links

Topics: Oscillazioni e moto armonico Concepts: Energia potenziale elastica · Energia cinetica · Conservazione dell’energia meccanica Skills: Conservazione dell’energia Objects: Molla