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← JEE Advanced 2020 · Paper 1
PhysicsMultiple correct+4 / -2Question 9 of 54

JEE Advanced 2020 · Paper 1, Physics question 9

Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity XX as follows: [position] =[Xα]= [X^{\alpha}]; [speed] =[Xβ]= [X^{\beta}]; [acceleration] =[Xp]= [X^{p}]; [linear momentum] =[Xq]= [X^{q}]; [force] =[Xr]= [X^{r}]. Then

Choose one or more options
Answer and worked solution

Official answer: A, B

Write [T]=Xt[T]=X^t and [M]=Xμ[M]=X^\mu, with [L]=Xα[L]=X^\alpha.

Exponents.

  • speed LT−1LT^{-1}: β=α−t\beta=\alpha-t.
  • acceleration LT−2LT^{-2}: p=α−2tp=\alpha-2t.
  • momentum MLT−1MLT^{-1}: q=μ+α−tq=\mu+\alpha-t.
  • force MLT−2MLT^{-2}: r=μ+α−2tr=\mu+\alpha-2t.

Checks.

  • (A) α+p=2α−2t=2β\alpha+p=2\alpha-2t=2\beta. True.
  • (B) p+q−r=(α−2t)+(μ+α−t)−(μ+α−2t)=α−t=βp+q-r=(\alpha-2t)+(\mu+\alpha-t)-(\mu+\alpha-2t)=\alpha-t=\beta. True.
  • (C) p−q+r=α−3tp-q+r=\alpha-3t, not α\alpha in general.
  • (D) p+q+r=2μ+3α−5tp+q+r=2\mu+3\alpha-5t, not β\beta in general.

(A), (B)

Topics:Dimensional analysis
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