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

The equation:

`a = r^nv^m`

The units for that equation must be:

`((m)/(s^2)) = (m)^(-1) ( (m)/(s) )^2`

Answer 2

Acceleration,
`a=k(v^2)/(r)`

Step-by-step 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(v^2)/(r)`

Hence, this is the required solution.