Word Count : 396

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हमारे पास अवमंदित आवर्त दोलक का समीकरण है: $$ m\frac{d^2x}{dt^2} + \gamma \frac{dx}{dt} + kx = 0 $$ दिए गए मान: $$ m = 0.25 \text{ kg}, \quad \gamma = 0.05 \text{ kg/s}, \quad k = 70 \text{ N/m} $$ हमें चाहिए: 1. दोलन काल $T_d$ 2. दोलन की वह संख्या जिसके बाद आयाम आधा रह जाता है 3. दोलन की वह संख्या जिसके बाद यांत्रिक ऊर्जा आधी रह जाती है --- ### Step 1: आवश्यक राशियों की गणना #### (a) Natural Frequency $$ \omega_0 = \sqrt{\frac{k}{m}} $$ $$ \omega_0 = \sqrt{\frac{70}{0.25}} = \sqrt{280} $$ $$ \omega_0 \approx 16.73 ________ _________ ____ ___ ___ _______ ______ _____ _______ _____ ____ _____.
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