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NEET
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NEET
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Physics
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Units and Dimensions

NEET · Physics · Units and Dimensions

A hypothetical system of units is created where the fundamental...

A hypothetical system of units is created where the fundamental unit of mass is α kg\alpha\text{ kg}, the fundamental unit of length is β m\beta\text{ m}, and the fundamental unit of time is γ s\gamma\text{ s}. The magnitude of a 1 N1\text{ N} force when expressed in this hypothetical system is :

Options

  1. Aαβγ−2\alpha\beta\gamma^{-2}
  2. Bαβγ2\dfrac{\alpha\beta}{\gamma^2}
  3. Cγ2αβ\dfrac{\gamma^2}{\alpha\beta}
  4. Dγαβ\dfrac{\gamma}{\alpha\beta}

Correct answer

C. γ2αβ\dfrac{\gamma^2}{\alpha\beta}

Step-by-step solution

The dimensional formula of force is [MLT−2][MLT^{-2}].

Let the SI system be system 1 and the hypothetical system be system 2.

Using the principle of conversion of units, n1[u1]=n2[u2]n_1[u_1] = n_2[u_2]:

n1[M1L1T1−2]=n2[M2L2T2−2]n_1 [M_1 L_1 T_1^{-2}] = n_2 [M_2 L_2 T_2^{-2}]

Given that n1=1n_1 = 1, M1=1 kgM_1 = 1\text{ kg}, L1=1 mL_1 = 1\text{ m}, T1=1 sT_1 = 1\text{ s} and M2=α kgM_2 = \alpha\text{ kg}, L2=β mL_2 = \beta\text{ m}, T2=γ sT_2 = \gamma\text{ s}.

1×[1 kg][1 m][1 s]−2=n2×[α kg][β m][γ s]−21 \times [1\text{ kg}] [1\text{ m}] [1\text{ s}]^{-2} = n_2 \times [\alpha\text{ kg}] [\beta\text{ m}] [\gamma\text{ s}]^{-2}

n2=1×(1α)×(1β)×(1γ)−2n_2 = 1 \times \left(\dfrac{1}{\alpha}\right) \times \left(\dfrac{1}{\beta}\right) \times \left(\dfrac{1}{\gamma}\right)^{-2}

n2=γ2αβn_2 = \dfrac{\gamma^2}{\alpha\beta}

Answer: γ2αβ\dfrac{\gamma^2}{\alpha\beta}
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