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 formula P = 2 1 k x 2 .
Multiply both sides by 2 to get 2 P = k x 2 .
Divide both sides by x 2 to isolate k .
The equivalent 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 want to find an equivalent equation solved for k .
Multiply both sides 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: 2 × P = 2 × 2 1 k x 2 2 P = k x 2
Divide both sides 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 for k So, the equation solved for k is: k = x 2 2 P
Final Answer Therefore, the correct equation Lupe should use 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 by controlling how much the suspension compresses under different loads. Similarly, in designing trampoline springs, knowing the spring constant helps determine the trampoline's bounce and weight capacity. In sports equipment, like bows and arrows, the spring constant of the bow determines the force and range of the arrow. These examples show how manipulating and understanding equations like P = 2 1 k x 2 can lead to practical and innovative solutions in engineering and design.
