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Chapter 6: Problem 55

A certain amount of work is required to stretch spring A a certain distance. Twice as much work is required to stretch spring B half that distance. Compare the spring constants of the two.

Short Answer

Expert verified
The spring constant of spring B is 8 times that of spring A.

Step by step solution

01

Understand the First Spring

Calculate the work done on spring A using the formula \(W = 0.5ka^2\), where 'k' is the spring constant and 'a' is the distance that spring is stretched. We can call this work as \(W_A\).
02

Understand the Second Spring

Calculate the work done on spring B using the formula \(W = 0.5kb^2\), where 'k' is the spring constant and 'b' (which is half of 'a') is the distance that spring B is stretched. Given that the work done on spring B is twice that done on spring A, we can write the equation as \(2W_A = 0.5kb^2\).
03

Compare the Spring Constants

Rearrange the formula of step 2 to find the spring constant 'k' for spring B in terms of spring A. Divide \(2W_A\) by \(0.5b^2\) to get \(k_B = 4W_A/b^2\). Substitute work done on spring A using formula from Step 1 as \(k_B = 4(0.5ka^2)/b^2 = 2ka^2/b^2\). Since 'b' is half of 'a', replace 'b' with 'a/2'. So, \(k_B = 2ka^2/(a/2)^2 = 2ka^2/0.25a^2=8k\). Thus, the spring constant of spring B is 8 times that of spring A.

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