Chapter 9, Problem 43A

Which requires more work: stretching a strong spring a certain distance or stretching a weak spring the same distance? Defend your answer.

To determine

To Choose:The spring which required more work for stretching.

Answer to Problem 43A

Workdone required to stretch strong spring is large than the work done required to stretchweak spring at same distance.

Explanation of Solution

Introduction:

Spring motion is one of the best examples of harmonic motion. Many of the systems in nature follow nature of spring system.

From kinematics theory, it is known that, for a given spring of mass m and spring constant k, force required to stretch a spring at distance x from its mean position is,

  $\therefore \text{F}\text{=}\text{-kx---------}\left(1\right)$

Negative sign indicate the direction of internal restoring force is opposite to the direction of displacement.

Then the work done to stretch a spring at distance x is,

  $\therefore \text{W}\text{=}\frac{\text{1}}{\text{2}}{\text{kx}}^{\text{2}}-----------\left(2\right)$

The magnitude of the work done can be visualized by the area under the curve of force-displacement graph as below:

From equation (1) if the spring is strong it requires more force to stretch at a distance x . Therefore, from equation (2) work done corresponding to this force is also large. The graphical representation shown in below diagram:

Similarly, for a weak spring, work done required to move spring at a distance x is:

Conclusion:

Therefore, from the above discussion, area under the curve for strong spring is greater than the area under the curve for weak spring. So, to move strong spring at a distance x requires more work done than the work done to move weak spring at same distance.