To solve the literal equation [tex]\( t = 2 \pi \sqrt{\frac{m}{k}} \)[/tex] for [tex]\( m \)[/tex], follow these steps:
1. Square both sides to eliminate the square root:
[tex]\[
t^2 = (2 \pi \sqrt{\frac{m}{k}})^2
\][/tex]
2. Simplify the right-hand side:
[tex]\[
t^2 = (2 \pi)^2 \left( \frac{m}{k} \right)
\][/tex]
Here, [tex]\((2 \pi)^2\)[/tex] simplifies to [tex]\(4 \pi^2\)[/tex]. So,
[tex]\[
t^2 = 4 \pi^2 \left( \frac{m}{k} \right)
\][/tex]
3. Multiply both sides by [tex]\( k \)[/tex] to isolate [tex]\( m \)[/tex] on the right-hand side:
[tex]\[
k t^2 = 4 \pi^2 m
\][/tex]
4. Divide both sides by [tex]\( 4 \pi^2 \)[/tex] to solve for [tex]\( m \)[/tex]:
[tex]\[
m = \frac{k t^2}{4 \pi^2}
\][/tex]
Thus, the solution for [tex]\( m \)[/tex] is:
[tex]\[
m = \frac{k t^2}{4 \pi^2}
\][/tex]
