Question

Consider a particle moving in a circular orbit of radius r with velocity v and acceleration a towards the centre of the orbit. Using dimensional analysis, show that a ∝ v^2 /r

Answer 1

we have to show, a ∝ v²/r using dimensional analysis,

here, a is acceleration towards the centre of the orbit.

so, dimension of a = [LT-²]

v is velocity so, dimension of v = [LT-¹]

and r is he radius of the circular orbit.

so, dimension of r = [L]

using dimensional analysis,

dimension of a ∝ (dimension of v)^x × (dimension of r)^y

⇒[LT-²] ∝ [LT-¹]^x × [L]^y

⇒[LT-²] ∝ [L]^(x + y) [T]^-x

on comparing both sides,

x + y = 1.....(1)

-2 = -x ⇒x = 2

so, y = -1 [ from equation (1) , ]

hence, x = 2 and y = -1

so, a ∝ v²r-¹ ⇒a ∝ v²/r