The acceleration (a) of a particle moving with uniform speed (v) in a circle of radius (r) is found to be propotional to r^n and v^m, where n and m are constants. Determine the values of n and m and write the simplest form of an equation of acceleration

m=2

n = - 1

a=Kv²/r

Explanation:

using dimensional analysis

V = meter/second = m/s

r = meter = m

a = meter/second²=m/s²

If a is proportional to r^n v^m

we have a=r^n v^m = (m) ^n (m/s) ^m = m/s²

from law of indices;

m^n+m/s^m = m/s²

using system of equations

n+m=1

m=2

so n = - 1

then a=kr^n v^m

a=Kv²/r

n=-1 and m=2

Explanation:

A particle is moving uniformly with acceleration "a"

Uniform speed is v

And radius of circle is r

The acceleration is proportional to

r^n and v^m

i. e

a∝ rⁿ

a ∝v^m

Combing the two

Then, a∝rⁿv^m

Let k be constant of proportionality

Then, a=krⁿv^m. Equation 1

So, we know that the centripetal acceleration keeping an object in circular path is given as

a=v²/r

Rearranging

a=v²r^-1. Equation 2

So comparing this to the proportional

Equating equation 1 and 2

krⁿv^m = v²r^-1

This shows that,

k=1

rⁿ = r^-1

Then, n = -1

Also, v^m = v²

Then, m=2

Therefore,

n=-1 and m=2