Answered

1. The centripetal force on a body of mass [tex]$m$[/tex] and velocity [tex]$v$[/tex] moving in a circular orbit of radius [tex][tex]$r$[/tex][/tex] is given by [tex]$F=\frac{mv^2}{r}$[/tex].

Write the dimensional formula of force.



Answer :

To determine the dimensional formula of the centripetal force given by the equation [tex]\( F = \frac{m v^2}{r} \)[/tex], we need to first determine the dimensional formulas of the variables involved: mass ([tex]\(m\)[/tex]), velocity ([tex]\(v\)[/tex]), and radius ([tex]\(r\)[/tex]).

1. Mass ([tex]\(m\)[/tex]):
- The dimensional formula for mass is: [tex]\([M]\)[/tex]

2. Velocity ([tex]\(v\)[/tex]):
- Velocity is defined as distance per unit time. Therefore, its dimensional formula is:
[tex]\[ [L][T]^{-1} \][/tex]

3. Radius ([tex]\(r\)[/tex]):
- Radius is a measure of length. Therefore, its dimensional formula is:
[tex]\[ [L] \][/tex]

Now, let's consider the given formula for the centripetal force:
[tex]\[ F = \frac{m v^2}{r} \][/tex]

First, find the dimensional formula for [tex]\( v^2 \)[/tex]:
[tex]\[ v^2 = \left( [L][T]^{-1} \right)^2 = [L]^2 [T]^{-2} \][/tex]

Next, substitute the dimensional formulas for [tex]\(m\)[/tex], [tex]\(v^2\)[/tex], and [tex]\(r\)[/tex] into the formula for force [tex]\(F\)[/tex]:
[tex]\[ F = \frac{m v^2}{r} = \frac{[M] \cdot [L]^2 [T]^{-2}}{[L]} \][/tex]

Simplify the expression by cancelling out the common terms:
[tex]\[ F = [M] \cdot [L]^2 \cdot [T]^{-2} \cdot [L]^{-1} = [M] \cdot [L]^{2-1} \cdot [T]^{-2} = [M] \cdot [L] \cdot [T]^{-2} \][/tex]

Therefore, the dimensional formula for force [tex]\(F\)[/tex] is:
[tex]\[ [M][L][T]^{-2} \][/tex]