Question

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 X as follows: [position] = [X^\(a\)]; [speed] = [X^β]; [acceleration] =[ X^p]; [linear momentum] = [X^q]; [force] = [X^r]. Then -

(A) \(a + p = 2β \)

(B) \(p + q - r\)=β 

(C)  \(p -q + r = a\)

 (D)\(p + q + r\)=β

Answer 1

Correct answer is (A,B)

Given L = x^\(a\) ……(1) 

LT^–1 = x^β ……(2) 

LT^–2 = x ^p ……(3) 

MLT^–1 = x ^q ……(4) 

MLT^–2 = x ^r ……(5)

From (4) M = x^q–β 

From (5) ⇒x ^q = x ^r x^q–β  

⇒ \(a\) + r – q = β ……(6) 

Replacing value '\(a\)' in equation (6) from (A) 

2β – p + r – q =β

p + q – r = β (B) 

Replacing value of 'β' in equation (6) from (A)

 2\(a\)+ 2r – 2q = \(a\) + p

\(a\) = p + 2q - 2r