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Suppose we are told that the acceleration a of a particle moving with uniform speed v in a circle of radius r is proportional to some power of r, say rn, and some power of v, say vm. Determine the values of n and m and write the simplest form of an equation for the acceleration.

Respuesta :

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The equation:
[tex]a = r^nv^m[/tex]

The units for that equation must be:
[tex](\frac{m}{s^2}) = (m)^{-1} ( \frac{m}{s} )^2[/tex]
aachen

Answer:

Acceleration, [tex]a=k\dfrac{v^2}{r}[/tex]

Explanation:

It is given that, the acceleration a of a particle moving with uniform speed v in a circle of radius r is proportional to some power of r and some power of v. Mathematically, it can be written as :

[tex]a\propto r^nv^m[/tex]

or

[tex]a=r^nv^m[/tex]...........(1)

Dimensional formula of a = [tex][LT^{-2}][/tex]

Dimensional formula of r = [tex][L][/tex]

Dimensional formula of v = [tex][LT^{-1}][/tex]

Using dimensional analysis in equation (1) as :

[tex][LT^{-2}]=[L]^n[LT^{-1}]^m[/tex]

[tex][LT^{-2}]=[L]^{n+m}[T^{-m}][/tex]

Equation both sides of equation as :

n + m = 1,  m = 2

This gives, n = -1

Use the value of m and n in equation (1) in order to get the formula :

[tex]a=kr^{-1}v^2[/tex]

[tex]a=k\dfrac{v^2}{r}[/tex]

Hence, this is the required solution.

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