Question

there are four springs stretched by the same mass. spring a stretches 25 cm. spring b stretches 10 cm. spring c stretches 100 cm. spring d stretches 1 cm. which spring has the largest spring constant? a b c d

Explanation:

Step1: Recall Hooke's Law

Hooke's Law is \(F = kx\), where \(F\) is the force applied (here \(F = mg\) as the mass \(m\) is the same for all springs, and \(g\) is the acceleration due to gravity), \(k\) is the spring constant, and \(x\) is the extension of the spring. We can rewrite the formula for \(k\) as \(k=\frac{F}{x}\). Since \(F = mg\) is constant (same mass \(m\) and \(g\) is a constant), \(k\propto\frac{1}{x}\).

Step2: Compare the extensions

We have \(x_A = 25\space cm\), \(x_B=10\space cm\), \(x_C = 100\space cm\), \(x_D = 1\space cm\).
Since \(k\propto\frac{1}{x}\), the spring with the smallest extension \(x\) will have the largest \(k\).

Answer:

D. Spring D