Physics, asked by rifatkhan1038, 5 hours ago

A particle of mass m is tied to a string and swung around in a circular path of radius r with a constant speed v. Derive a formula for the centripetal force F exerted by the particle on our hand using the method of dimension

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Answered by RISH4BH
134

Need to FinD :-

  • The expression of centripetal force F exerted by the particle on our hand using the method of dimension .

Given that , force acting on a particle depends upon mass of the particle , radius of the track and the velocity of the particle . And we need to find the expression of the force .

Let the expression ,

\sf\longrightarrow Force = a [ m^x v^y r^z ] \\\\\\\sf\longrightarrow Force =[m^x] [v^y ] [r^z] </p><p>

where ,

  • a is dimensionless constant.
  • x , y and z are the powers on m , v and r respectively .

We know that the Dimensional Formula of Force is ,

\sf\longrightarrow\red{ Force = [ M^1 L^1 T^{-2}] }

Now , by the Principal of Homogeneity ,

\sf\longrightarrow [ M^1 L^1 T^{-2}] = [m^x] [v^y ] [r^z]  \\\\\\\sf\longrightarrow [M^1L^1T^{-2}] = [ M]^x [ LT^{-1}]^y [L]^z  \\\\\\\sf\longrightarrow  [M^1L^1T^{-2}] = [ M^x L^{(y+z)} T^{-y} ]

On comparing ,

\sf\longrightarrow x = 1\\\\

\sf\longrightarrow y + z = 1 \\\\

\sf\longrightarrow -y = -2

On solving ,

\longrightarrow\textsf{\textbf{ x = 1}}\\\\

\longrightarrow\textsf{\textbf{ y = 2}}\\\\

\longrightarrow\textsf{\textbf{ z= -1}}

Therefore , the expression for the Force is ,

\sf\longrightarrow Force = a [ m^x v^y r^z ] \\\\\\\sf\longrightarrow Force = a [ m^1 v^2 r^{-1} ] \\\\\\\sf\longrightarrow Force = 1 [ m^1 v^2 r^{-1} ] \qquad \bigg\{ \red{\sf The \ Dimensionless \ constant \ here \ is \ 1 .}\bigg\} \\\\\\\sf\longrightarrow  \underset{\blue{\sf Centripetal\ Force }}{\underbrace{\boxed{\pink{\frak{ Force_{(Centripetal)}  =\dfrac{mv^2}{r}}}}}}

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