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]
