Question

Don't PosT IrreŁevanT Answer❎
Hello❗
♦Centripetal Force 'F' acting on a particle of mass 'm' moving in a circle of radius 'r' with velocity 'v' depends on m, v and r. Find The expression for F.♦​

Answer 1

Given :

▪ A particle of mass 'm' is moving in a circular path of radius 'r' with velocity 'v'.

▪ Centripetal force acting on particle = F

To Find :

▪ Relation between F, m, v and r.

SolutioN :

⚾ Dimension formulas :-

$\star\sf\:F=[M^1L^1T^{-2}]\\ \\ \star\sf\:m=[M^1L^0T^0]\\ \\ \star\sf\:v=[M^0L^1T^{-1}]\\ \\ \star\sf\:r=[M^0L^1T^0]$

ATQ,

$\dashrightarrow\bf\:F=m^av^br^c\\ \\ \dashrightarrow\sf\:[M^1L^1T^{-2}]=[M^1L^0T^0]^a[M^0L^1T^{-1}]^b[M^0L^1T^0]^c\\ \\ \dashrightarrow\sf\:[M^1L^1T^{-2}]=[M^a][L^bT^{-b}][L^c]\\ \\ \dashrightarrow\sf\:[M^1L^1T^{-2}]=[M^aL^{b+c}T^{-b}]\\ \\ \Rightarrow\bf\:a=1\\ \\ \Rightarrow\bf\:b=2\\ \\ \Rightarrow\bf\:b+c=1\rightarrow\:c=-1\\ \\ \mapsto\sf\:F=m^1v^2r^{-1}\\ \\ \mapsto\underline{\boxed{\bf{\purple{F=\dfrac{mv^2}{r}}}}}\:\gray{\bigstar}$