The centripetal force F acting on a particle moving uniformly in a circle may depend upon mass m, velocity v and radius r of the circle. Derive the formula for F using the method of dimensions.

Question

The centripetal force F acting on a particle moving uniformly in a circle may depend upon mass m, velocity v and radius r of the circle. Derive the formula for F using the method of dimensions.

Infinity Learn · Accepted Answer

Let $$F=k(m)x(v)y(r)z$$… (i) Here, k is a dimensionless constant of proportionality. Writing the dimensions of RHS and LHS in Eq. (i), we $$haveMLT-2=$$[M]$$xLT-1y$$[ L]$$z=Mx$$ $$Ly+2$$ $$T-yEquating$$ the powers of M, L and T of both sides, we have, $$x=1$$,$$y=2$$  and  $$y+z=1$$  or  $$z=1-y=-1Putting$$ the values in Eq. (i), we $$getF=kmv2r-1=kmv2rorF=mv2r$$ (where$$, $$k=1$$ $$)

The centripetal force $F$ acting on a particle moving uniformly in a circle may depend upon mass $m$ , velocity $v$ and radius $r$ of the circle. Derive the formula for $F$ using the method of dimensions.

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