The potential energy, [tex]P[/tex], in a spring is represented using the formula [tex]P=\frac{1}{2} k x^2[/tex]. Lupe uses an equivalent equation, which is solved for [tex]k[/tex], to determine the answers to her homework.
Which equation should she use?
[tex]k=2 P x^2[/tex]
[tex]k=\frac{1}{2} P x^2[/tex]
[tex]k=\frac{2 P}{x^2}[/tex]
[tex]k=\frac{p}{2 x^2}[/tex]
Answer (1)
Start with the given equation: P = 2 1 k x 2 .
Multiply both sides by 2: 2 P = k x 2 .
Divide both sides by x 2 : x 2 2 P = k .
The equation solved for k is: k = x 2 2 P .
Explanation
Understanding the Problem We are given the formula for potential energy in a spring: P = 2 1 k x 2 , where P is the potential energy, k is the spring constant, and x is the displacement. We need to find an equivalent equation solved for k .
Multiply by 2 To solve for k , we need to isolate k on one side of the equation. We can do this by multiplying both sides of the equation by 2: 2 P = k x 2
Divide by x^2 Next, we divide both sides of the equation by x 2 :
x 2 2 P = k
Final Equation Therefore, the equation solved for k is: k = x 2 2 P
Examples
Understanding the relationship between potential energy, spring constant, and displacement is crucial in various fields, such as mechanical engineering and physics. For instance, when designing a suspension system for a car, engineers use this formula to determine the appropriate spring constant needed to provide a comfortable ride. By knowing the potential energy the spring needs to absorb and the maximum displacement it should undergo, they can calculate the required spring constant, ensuring the suspension system performs optimally.
