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Chapter 4: Problem 57

Two springs have force constants, \(K_{1}\) and \(K_{2}\left(K_{1}>K_{2}\right)\) The work done, when both are stretched by the same amount of length will be (A) Equal (B) Greater for \(K_{1}\) (C) Greater for \(K_{2}\) (D) Given data is incomplete

Short Answer

Expert verified
(B) Greater for \(K_1\).

Step by step solution

01

Identify the given information

We are given the following information: - Two springs have force constants \(K_1\) and \(K_2\) (\(K_1 > K_2\)). - Both springs are stretched by the same amount of length.
02

Write down the formula for the work done on a spring

The formula for the work done on a spring is given by: \[W = \frac{1}{2}kx^2\]
03

Calculate the work done for each spring

Using the given formula, we can calculate the work done on each spring. For Spring 1: \[W_1 = \frac{1}{2}K_1x^2\] For Spring 2: \[W_2 = \frac{1}{2}K_2x^2\]
04

Compare the work done for each spring

Now we need to compare \(W_1\) and \(W_2\) to determine which one is greater. Since \(K_1 > K_2\), it is clear that \(\frac{1}{2}K_1x^2 > \frac{1}{2}K_2x^2\), or in other words: \[W_1 > W_2\]
05

Choose the correct answer

Comparing the work done on both springs, we can see that the work done is greater for the spring with a larger force constant, \(K_1\). Therefore, the correct answer is: (B) Greater for \(K_1\).

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