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 (2)
Start with the formula: 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 = f r a c 1 2 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 .
Multiplying by 2 To isolate k , we need to get rid of the f r a c 1 2 and the x 2 terms. First, let's multiply both sides of the equation by 2 to eliminate the fraction: 2 × P = 2 × 2 1 k x 2 2 P = k x 2
Dividing by x 2 Now, we need to isolate k by dividing both sides of the equation by x 2 :
x 2 2 P = x 2 k x 2 x 2 2 P = k
Final Equation So, the equation solved for k is: k = x 2 2 P
Examples
Understanding the spring constant is crucial in various real-world applications. For instance, when designing suspension systems for vehicles, engineers use the spring constant to ensure a comfortable ride and optimal handling. Similarly, in the construction of buildings, the spring constant of materials helps in assessing their ability to withstand loads and vibrations, ensuring structural integrity. Even in sports equipment like trampolines, the spring constant determines the bounce and responsiveness, affecting performance and safety.
To solve for the spring constant k in the equation for potential energy, the correct expression is k = x 2 2 P . This is derived by multiplying the original equation by 2 and then dividing by x 2 . Thus, Lupe should use the option ' k = x 2 2 P '.
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