Suppose, we are told that the acceleration of a particle moving in a circle of radius r with uniform speed v is proportional to some power of r, say `r^n`, and some power of v, say `v^m` , determine the powers of r and v ?

(Ans: n = -1, m = 2)

Given:

Acceleration (a) is proportional to `r^n`
Acceleration (a) is proportional to `v^m`

To Find:

To determine the values of the power of r and v  


Solution:

According to the given conditions, the formula for the speed (v) of sound will be

a ∝ `r^n` `v^m`

or

a = Constant `r^n` `v^m`  ------Equation (a)

Now using dimensions methods we will find the values of power n and m.

R.H.S. Dimensions of the equation (1):

[a] = [LT⁻²]

L.H.S. Dimensions of the equation (1):

[r] = [L]

[v] = [LT⁻¹] 
 

Now writing the dimension of both sides of the equation (1) we get

[LT⁻²] = `[L]^n` `[LT⁻¹]^m`

By separating each dimension quantity both side

[L] [T]⁻² = `[L]^{n+m}` `[T]^{-m}`

Now equating the power (exponent) of [L] and [T] on both sides, we get

By equating the powers (exponents) of  [L]
1 = n + m    ----- eqn (i)

By equating the exponent of  [T]
-2 = -m    ----- eqn (ii)

From eqn (ii) we get

m = 2 --------Ans(1)

and substituting the value of m in eqn (i)  we get

1 = n + 2

n = -1 --------Ans(2)

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Numerical Problem 1.10
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List of 11th Physics Numerical Problems of Chapter - 1
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