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] = [𝑋𝛼]; [speed] = [𝑋𝛽]; [acceleration] =[𝑋𝑝]; [linear momentum] = [𝑋𝑞]; [force] = [𝑋𝑟]. Then

Answer 1

Position = [x𝛼]

 

⇒ [L] = [𝑋𝛼] ..... (1)

 

speed = x𝛽

 

LT-1 = 𝑋𝛽

 

𝑋𝛼 T-1 = X𝛽 (from (1)

 

T-1 = 𝑋𝛼-𝛽….. (2)

 

Acceleration = XP

 

LT-1 × T-1 = Xp

 

𝑋𝛽.𝑋𝛼-𝛽 = xp

 

𝑋2𝛼-𝛽 = xp

 

⇒ 2β – α = p

 

α + p = 2β

 

Option (a)

 

[Linear momentum] = xq

 

M LT-1 = xq

 

M.X 𝛽 = xq

 

M = xq-𝛽 …... (3)

 

[force] = xr

 

M LT-2 = xr

 

M.Xp = xr

 

M = xr-p …..(4)

 

From (3) and (4), q - β = r - p

 

p + q - r = β option (B)

 

From (a) p + q -r = α / 2 + p / 2

 

P / 2 + q - r = α / 2

 

 

The answer is a)α + p = 2β and b) P / 2 + q - r = α / 2