Question
(क) वक्रों और
से परिबद्ध और
के घनत्व वाले एक पतली शीट का गुरुत्व केन्द्र ज्ञात कीजिए।
(ख) और
से परिबद्ध ठोस घनाकृति का द्रव्यमान ज्ञात कीजिए, जबकि घनत्व फलन,
हो।
Answer :
Word Count : 399
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(क) वक्र (y = 4x^2) और रेखा (x = 4) के बीच घनत्व (\delta(x, y) = y) वाली पतली शीट का गुरुत्वकेंद्र ज्ञात करें। सबसे पहले, मास (M) ज्ञात करें: [ M = \int_0^4 \delta(x, y) , ds ] जहाँ (ds = \sqrt{1 + (y')^2} , dx), और (y' = \frac{dy}{dx} = 8x)। [ ds = \sqrt{1 + (8x)^2} , dx = \sqrt{1 + 64x^2} , dx ] (\delta(x, y) = y = 4x^2) के लिए: [ M = \int_0^4 4x^2 \sqrt{1 + 64x^2} , dx ] गुरुत्वकेंद्र के निर्देशांक: [ \bar{x} = \frac{1}{M} \int x , \delta(x, y) , ds = \frac{\int_0^4 x \cdot 4x^2 \sqrt{1 + 64x^2} , dx}{\int_0^4 4x^2 \sqrt{1 + 64x^2} , dx} = \frac{\int_0^4 4x^3 \sqrt{1 + 64x^2} ,
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