Question

The amplitude of a lightly damped oscillator decreases by 3.42% during each cycle. What percentage of the mechanical energy of the oscillator is lost in each cycle?

Answer 1

The percentage of the mechanical energy of the oscillator lost in each cycle is 6.72%

Step-by-step explanation:

Mechanical energy (Potential energy, PE) of the oscillator is calculated as;

PE = ¹/₂KA²

During the first oscillation;

PE₁ = ¹/₂KA₁²

During the second oscillation;

A₂ = A₁ - 0.0342A₁ = 0.9658A₁

PE₂ = ¹/₂KA₂²

PE₂ = ¹/₂K (0.9658A₁)²

PE₂ = (0.9658²)¹/₂KA₁²

PE₂ = (0.9328)¹/₂KA₁²

PE₂ = 0.9328PE₁

Percentage of the mechanical energy of the oscillator lost in each cycle;

`Change \ in \ percent= (PE_2 - PE_1)/(PE_1) \\\\Change \ in \ percent= (0.9328PE_1 -PE_1)/(PE_1) *100\\\\Change \ in \ percent= (-0.0672PE_1)/(PE_1)*100 \\\\Change \ in \ percent= -0.0672*100\\\\Change \ in \ percent= -6.72 \ \% \\\\Loss \ in \ percent= 6.72 \ \%`

Therefore, the percentage of the mechanical energy of the oscillator lost in each cycle is 6.72%