Which equation should she use?
A. [tex]$k=2 P x^2$[/tex]
B. [tex]$k=\frac{1}{2} P x^2$[/tex]
C. [tex]$k=\frac{2 P}{x^2}$[/tex]
D. [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 : k = x 2 2 P .
The equation Lupe should use 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 x 2 terms. First, let's multiply both sides of the equation by 2: 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
The Result So, the equation for k in terms of P and x is: k = x 2 2 P
Examples
Understanding the spring constant is crucial in designing suspension systems for vehicles. If you know the potential energy a spring needs to store and the maximum compression distance, you can calculate the required spring constant. This ensures the suspension provides a comfortable ride and can handle various loads without bottoming out. Similarly, in designing catapults or other mechanical devices that use springs, knowing how to calculate the spring constant helps in achieving the desired performance and efficiency.
The correct equation for the spring constant k in terms of potential energy P and displacement x is k = x 2 2 P , which corresponds to option C. This formula is derived from the potential energy equation for springs. Therefore, the chosen option is C: k = x 2 2 P .
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