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The centripetal force F acting on a body moving uniformly in a circle may depend upon its mass m velocity v and radius r of the circle. Derive the formula for centripetal force using the method of dimension
Carries 3 Marks

Class 11 CBSE

Answer 1

Explanation:

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Answer 2

Solution :

Force = F

Centripetal Force = $C_{f}$

Dimensional formula of Centripetal Force : [ M¹ L¹ $T^{-2}$ ]

Mass = m (in kg)

Velocity = v (in m/sec)

Radius = r (in m)

$C_{f}\: = \:\dfrac{mv^{2}}{r}$

$C_{f}\: \propto m^{a}.v^{b}.r^{c}$

As per dimensional formula

For mass m = [M¹]

For velocity v = $[L^{1} T^{-1}]$

[L is radius , T = Times]

For radius r = [L¹]

[ M¹ L¹ $T^{-2}$ ] =$[M^{1}]^{a}.[L^{1} T^{-1}]^{b}.[L^{1}]^{c}$

[ M¹ L¹ $T^{-2}$ ] = $[M]^a.[L]^{b+c}.[T]^{-b}$

Now Compare the powers of same base

a = 1 {for M}

-b = -2 or b = 2 {for T}

b+c = 1 {for L}

2+c = 1 or c = -1

Put the values in a , b , c

$C_{f}\: =\: k.m^{1}.v^{2}.r^{-1}$

$C_{f}\: =\: k.\dfrac{mv^{2}}{r}$

k is constant in the equation to remove proportionality symbol

$\boxed{C_{f}\: =\: k.\dfrac{mv^{2}}{r}}$