Physics Q92 – AP-EAMCET 2026

The mechanical energy of a damped harmonic oscillator at a time

t = 0

80 J

. If its mechanical energy becomes

40 J

after completing

15

oscillations, then the extra oscillations the oscillator has to complete so that its mechanical energy becomes

10 J

Clue 1 of 3: Energy Decay in Damped Oscillators

The mechanical energy of a damped harmonic oscillator decreases exponentially over time (or number of oscillations) due to dissipative forces like air resistance or friction.

🧠 Full Solution Path

Step 1: Understand the Energy Decay Model✦ Active

The mechanical energy of a damped harmonic oscillator decays exponentially with the number of oscillations. The formula governing this decay is $E_{N} = E_{0} e^{- k N}$ , where $E_{N}$ is the energy after $N$ oscillations, $E_{0}$ is the initial energy, and $k$ is a damping constant.

Step 2: Calculate the Damping Constant (

k

)○ Expand

Given: initial energy $E_{0} = 80 J$ at $N = 0$ . After $N_{1} = 15$ oscillations, the energy $E_{1} = 40 J$ . Substitute these values into the energy decay formula:

40 = 80 e^{- k \times 15}

\frac{40}{80} = e^{- 15 k}

0.5 = e^{- 15 k}

Take the natural logarithm of both sides:

\ln (0.5) = - 15 k

- \ln (2) = - 15 k

k = \frac{\ln (2)}{15}

💡 Teacher's Secret Hint

Remember that $\ln (0.5) = \ln (1 / 2) = - \ln (2)$ . This simplification helps avoid negative signs in calculations.

Step 3: Calculate Total Oscillations (

N_{2}

) for the Target Energy○ Expand

The target energy $E_{2} = 10 J$ . We need to find the total number of oscillations $N_{2}$ required for the energy to reach $10 J$ :

10 = 80 e^{- k N_{2}}

\frac{10}{80} = e^{- k N_{2}}

0.125 = e^{- k N_{2}}

\frac{1}{8} = e^{- k N_{2}}

Take the natural logarithm of both sides:

\ln (\frac{1}{8}) = - k N_{2}

- \ln (8) = - k N_{2}

\ln (8) = k N_{2}

Since $\ln (8) = \ln (2^{3}) = 3 \ln (2)$ , we have:

3 \ln (2) = k N_{2}

Substitute the value of $k = \frac{\ln (2)}{15}$ from Step 2:

3 \ln (2) = (\frac{\ln (2)}{15}) N_{2}

Solving for $N_{2}$ :

N_{2} = 3 \times 15 = 45 oscillations

💡 Teacher's Secret Hint

Simplify logarithmic terms like $\ln (1 / 8)$ to $- \ln (8)$ and $\ln (8)$ to $3 \ln (2)$ to make the algebra easier and cancel out common terms like $\ln (2)$ .

Step 4: Determine the Extra Oscillations○ Expand

The oscillator has already completed $N_{1} = 15$ oscillations. The total oscillations required for the energy to become $10 J$ is $N_{2} = 45$ . The extra oscillations needed are:

Extra oscillations = N_{2} - N_{1} = 45 - 15 = 30 oscillations

💡 Teacher's Secret Hint

Always double-check what the question is asking for: total oscillations, or *extra* oscillations after a certain point. This is a common point of error.

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.

Physics Question 92 – AP-EAMCET 2026

Clue 1 of 3: Energy Decay in Damped Oscillators

🧠 Full Solution Path