Question

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.

Answer 1

Answer:

Acceleration, $a=k\dfrac{v^2}{r}$

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 :

$a\propto r^nv^m$

or

$a=r^nv^m$...........(1)

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

Dimensional formula of r = $[L]$

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

Using dimensional analysis in equation (1) as :

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

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

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 :

$a=kr^{-1}v^2$

$a=k\dfrac{v^2}{r}$

Hence, this is the required solution.

Answer 2

The equation:
$a = r^nv^m$

The units for that equation must be:
$(\frac{m}{s^2}) = (m)^{-1} ( \frac{m}{s} )^2$