Certainly! Let's find the steps required to solve the equation [tex]\( t = 2 \pi \sqrt{\frac{m}{k}} \)[/tex] for [tex]\( m \)[/tex].
### Step-by-Step Solution:
1. Start with the given equation:
[tex]\[
t = 2 \pi \sqrt{\frac{m}{k}}
\][/tex]
2. Square both sides of the equation to eliminate the square root. Squaring [tex]\( t \)[/tex] and [tex]\( 2 \pi \sqrt{\frac{m}{k}} \)[/tex] gives:
[tex]\[
t^2 = \left(2 \pi \sqrt{\frac{m}{k}}\right)^2
\][/tex]
3. Simplify the right-hand side, noting that squaring [tex]\( 2 \pi \)[/tex] results in [tex]\( 4 \pi^2 \)[/tex] and squaring [tex]\( \sqrt{\frac{m}{k}} \)[/tex] results in [tex]\( \frac{m}{k} \)[/tex]:
[tex]\[
t^2 = 4 \pi^2 \left(\frac{m}{k}\right)
\][/tex]
4. Isolate [tex]\( m \)[/tex] by multiplying both sides of the equation by [tex]\( k \)[/tex] to move [tex]\( \frac{1}{k} \)[/tex] on the right-hand side out of the fraction:
[tex]\[
t^2 k = 4 \pi^2 m
\][/tex]
5. Solve for [tex]\( m \)[/tex] by dividing both sides of the equation by [tex]\( 4 \pi^2 \)[/tex]:
[tex]\[
m = \frac{k t^2}{4 \pi^2}
\][/tex]
So, the literal equation solved for [tex]\( m \)[/tex] is:
[tex]\[
m = \frac{k t^2}{4 \pi^2}
\][/tex]
This completes our step-by-step solution to isolate [tex]\( m \)[/tex] in the given equation.
