To solve the given equation [tex]\( t = 2 \pi \sqrt{\frac{m}{k}} \)[/tex] for [tex]\( m \)[/tex], follow these steps:
1. Isolate the square root term:
Start by isolating the term involving [tex]\( m \)[/tex]. Divide both sides of the equation by [tex]\( 2 \pi \)[/tex]:
[tex]\[
\frac{t}{2 \pi} = \sqrt{\frac{m}{k}}
\][/tex]
2. Eliminate the square root:
To remove the square root, square both sides of the equation:
[tex]\[
\left(\frac{t}{2 \pi}\right)^2 = \left(\sqrt{\frac{m}{k}}\right)^2
\][/tex]
This simplifies to:
[tex]\[
\frac{t^2}{(2 \pi)^2} = \frac{m}{k}
\][/tex]
3. Solve for [tex]\( m \)[/tex]:
Isolate [tex]\( m \)[/tex] by multiplying both sides of the equation by [tex]\( k \)[/tex]:
[tex]\[
m = k \cdot \frac{t^2}{(2 \pi)^2}
\][/tex]
4. Simplify the solution:
Simplify the denominator:
[tex]\[
(2 \pi)^2 = 4 \pi^2
\][/tex]
Thus, the equation becomes:
[tex]\[
m = \frac{k t^2}{4 \pi^2}
\][/tex]
Therefore, the solution to the equation [tex]\( t = 2 \pi \sqrt{\frac{m}{k}} \)[/tex] for [tex]\( m \)[/tex] is:
[tex]\[
m = \frac{k t^2}{4 \pi^2}
\][/tex]
