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प्रश्न
v = `at + b/(t + c) + "v"_0` is a dimensionally valid equation. Obtain the dimensional formula for a, b and c where v is velocity, t is time and v0 is initial velocity.
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उत्तर
Given: v = `at + b/(t + c) + "v"_0`
As only dimensionally identical quantities can be added together or subtracted from each other, each term on R.H.S. has dimensions of L.H.S. i.e., dimensions of velocity.
∴ [L.H.S.] = [v] = [L1T−1]
This means, [at] = [v] = [L1T−1]
Given, t = time has dimension [T1]
∴ [a] = `(["L"^1"T"^-1])/(["t"]) = (["L"^1"T"^-1])/(["T"^1]) = ["L"^1"T"^-2] = ["L"^1"M"^0"T"^-2]`
Similarly,
[c] = [t] = [T1] = [L0M0T1]
∴ `(["b"])/(["T"^1]) = ["v"] = ["L"^1"T"^-1]`
∴ [b] = [L1T-1] × [T1] = [L1] = [L1M0T0]
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