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This is known as principle of homogeneity. If x = At + Bt2 where x = length and t = time, then find the dimensional formula of [A] and [B].

Physics

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Answer

[A]=[LT−1][A] = [L T^{-1}]

[B]=[LT−2][B] = [L T^{-2}]

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दिए गए समीकरण x=At+Bt2x = At + Bt^2 में, हमें AA और BB के विमीय सूत्र (dimensional formula) ज्ञात करने हैं। यह विमीय समरूपता के सिद्धांत (principle of homogeneity) पर आधारित है, जिसके अनुसार एक समीकरण के सभी पदों के आयाम समान होने चाहिए।

दिया गया है:
xx एक लंबाई है, इसलिए [x]=[L][x] = [L]
tt एक समय है, इसलिए [t]=[T][t] = [T]

  1. AA का विमीय सूत्र ज्ञात करें।

    विमीय समरूपता के सिद्धांत के अनुसार, xx का आयाम AtAt के आयाम के बराबर होना चाहिए:

    [x]=[At][x] = [At] [L]=[A][T][L] = [A][T]

    AA के लिए हल करने पर:

    [A]=[L][T][A] = \frac{[L]}{[T]} [A]=[LT−1][A] = [L T^{-1}]

  2. BB का विमीय सूत्र ज्ञात करें।

    विमीय समरूपता के सिद्धांत के अनुसार, xx का आयाम Bt2Bt^2 के आयाम के बराबर होना चाहिए:

    [x]=[Bt2][x] = [Bt^2] [L]=[B][T2][L] = [B][T^2]

    BB के लिए हल करने पर:

    [B]=[L][T2][B] = \frac{[L]}{[T^2]} [B]=[LT−2][B] = [L T^{-2}]

    अतः, AA और BB के विमीय सूत्र हैं:

    [A]=[LT−1]\boxed{[A] = [L T^{-1}]} [B]=[LT−2]\boxed{[B] = [L T^{-2}]}
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