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 smallest spring constant? a b c d

Explanation:

Step1: Recall Hooke's Law

Hooke's Law states that \( F = kx \), where \( F \) is the force applied (here, the weight of the mass, so \( F \) is the same for all springs), \( k \) is the spring constant, and \( x \) is the displacement (stretch) of the spring. Rearranging for \( k \), we get \( k=\frac{F}{x} \).

Step2: Analyze the relationship

Since \( F \) is constant (same mass, so same weight), \( k \) is inversely proportional to \( x \) ( \( k \propto \frac{1}{x} \) ). This means the larger the stretch \( x \), the smaller the spring constant \( k \).

Step3: Compare the stretches

The stretches are: Spring A: \( 25 \, \text{cm} \), Spring B: \( 10 \, \text{cm} \), Spring C: \( 100 \, \text{cm} \), Spring D: \( 1 \, \text{cm} \). Among these, \( 100 \, \text{cm} \) (Spring C) is the largest stretch.

Step4: Determine the smallest \( k \)

Since \( k \) is inversely proportional to \( x \), the spring with the largest \( x \) (Spring C) will have the smallest \( k \).

Answer:

C. Spring C