Explanation Given: The work done to stretch the spring from 10 cm to 12 cm is 6 J. The work done to stretch the spring from 12 cm to 14 cm is 10 J. Hook’sLaw: The force required to maintain a spring stretched x units beyond its natural length is proportional to x : f ( x ) = k x (1) Here, k is called spring constant. Calculation: Consider the original length of the spring as L cm. The work done to stretch the spring from 10 cm to 12 cm is 6 J. Initially, the length of the stretched spring is 10 cm. Take the change in length of the spring a as shown below: a = ( 10 − L ) cm = ( 10 − L ) cm × 1 m 100 cm = ( .10 − 0.01 L ) m . A work of 6 J is done on this spring This stretched spring is stretched further upto a length of 12 cm. Take the change in length of the spring b as shown below: b = ( 12 − L ) cm = = ( 12 − L ) cm × 1 m 100 cm = ( 0.12 − 0.01 L ) m . Show the expression for work done to to stretch the spring from a to b , when a force f ( x ) is applied on it. W = lim n → ∞ ∑ i = 1 n f ( x i ∗ ) Δ x = ∫ a b f ( x ) d x (2) Modify Equation (2) using Equation (1). W = ∫ a b k x d x (3) Substitute ( 0.10 − 0.01 L ) m for a , ( 0.12 − 0.01 L ) m for b , 6 J for W in Equation (3). 6 = ∫ 0.10 − 0.01 L 0.12 − 0.01 L k x d x = k [ x 2 2 ] 0.10 − 0.01 L 0.12 − 0.01 L = k 2 × [ ( 0.12 − 0.01 L ) 2 − ( 0.10 − 0.01 L ) 2 ] 12 = k × [ ( 0.12 − 0.01 L ) 2 − ( 0.10 − 0.01 L ) 2 ] (4) Show the mathematical formula. a 2 − b 2 = ( a + b ) ( a − b ) (5) Apply Equation (5) in Equation (4). 12 = k × [ ( 0.12 − 0.01 L + 0.10 − 0.01 L ) ( 0.12 − 0.01 L − 0.10 + 0.01 L ) ] = k × ( 0.22 − 0.02 L ) ( 0.02 ) = k × ( 0.22 − 0.02 L ) ( 0.02 ) (6) The work done to stretch the same spring from 12 cm to 14 cm is 10 J

8th Edition

ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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Chapter 5.4, Problem 12E

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To find: The natural length of the spring.

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The natural length of the spring.

Solution Summary: The author explains how the force required to maintain a spring stretched beyond its natural length is proportional to x.

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To find: The natural length of the spring.

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