Question

the potential energy, p, in a spring is represented using the formula ( p = \frac{1}{2}kx^2 ). lupe uses an equivalent equation, which is solved for ( k ), to determine the answers to her homework. which equation should she use? ( \bigcirc k = 2px^2 ) ( \bigcirc k = \frac{1}{2}px^2 ) ( \bigcirc k = \frac{2p}{x^2} ) ( \bigcirc k = \frac{p}{2x^2} )

Explanation:

Step1: Start with the given formula

We have the formula for the potential energy in a spring: \( P=\frac{1}{2}kx^{2} \). We need to solve this formula for \( k \).

Step2: Multiply both sides by 2

To eliminate the fraction, we multiply both sides of the equation by 2.
\( 2\times P = 2\times\frac{1}{2}kx^{2} \)
Simplifying the right - hand side, the 2 and \( \frac{1}{2} \) cancel out, and we get \( 2P=kx^{2} \).

Step3: Divide both sides by \( x^{2} \)

To isolate \( k \), we divide both sides of the equation \( 2P = kx^{2} \) by \( x^{2} \) (assuming \( x
eq0 \)).
\( \frac{2P}{x^{2}}=\frac{kx^{2}}{x^{2}} \)
Since \( \frac{kx^{2}}{x^{2}} = k \) (for \( x
eq0 \)), we have \( k = \frac{2P}{x^{2}} \).

Answer:

\( k=\frac{2P}{x^{2}} \) (the third option)