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If momentum (p), area (A) and time(t) ar...

If momentum `(p)`, area `(A)` and time`(t) `are taken to be fundamental quantities then energy has the dimensional formula

A

`[p^(1)A^(-1)t^(-1)]`

B

`[p^(2)A^(1)t^(1)]`

C

`[p^(1)A^(1//2)t^(1)]`

D

`[p^(1)A^(1//2)t^(-1)]`

Text Solution

Verified by Experts

The correct Answer is:
D
Let energy `E = kp^(a) A^(b)t^(c )`…(i)
where is k a dimensionless constant proportionality equating dimension an both sides of(i) we get
`[ML^(2)T^(-2)] = [MLT^(-1)]^(a) [M^(0)L^(2)T^(0)]^(b) [M^(0)L(0)T]^(c)`
`[L]= [M^(a)L^(a+2b)T^(a+c)]`
Applying the principle of homogeneity of dimensions we get
`a = 1`...(ii)
`a + 2b = 2`...(iii)
`-a+c = -2`...(iv)
On solving eqs `(ii),(iii)` and `(iv)` we get
`a = 1, b = (1)/(2) , c=-1`
`:. [E] = [p^(1)A^(1//2)c^(-2)]`
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Knowledge Check

  • If momentum (P) , area (A) and time (7) are taken to be fundamental quntities, then power has the dimensional formula.

    A
    `(P^(1)A^(-1)T^(1))`
    B
    `(P^(2)A^(1)T^(1))`
    C
    `(P^(1)A^(-1//2)T^(1))`
    D
    `(P^(1)A^(1//2)T^(-2))`

  • If momentum (p), Area (A) and time (T) are takes as fundamental quantities, then energy has the dimensional formula :

    A
    `[pA^(1//2)T^(-1)]`
    B
    `[pA^(-1//2)T^(1)]`
    C
    `[p^(2)AT]`
    D
    `[pA^(-1)T]`

  • If momentum P, area A and time T are selected as fundamental quantities then energy has the dimensional formula

    A
    `[P^(2)A^(2)T^(-1)]`
    B
    `[P^(-1)A^(-1//2)T^(1)]`
    C
    `[P^(1)A^(1//2)T^(-1)]`
    D
    `[P^(-2)A^(-2)T^(1)]`

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