Suppose I have a spring of spring constant 150 N/m.One person pulls with it a force of 15 N. The extension produced is 0.1 m. Now another person comes and pulls with a force of 30 N (The first person is still there). The final extension is 0.3 m.The initial and final potential energies of the spring are 0.75 J and 6.75 J respectively. The work done by the bigger force is 6 J which is exactly equal to the change in P.E of the spring. So where did the work done by the smaller force (when they were pulling it together) go? Why didn't it help to further increase the potential energy of the spring?
[Physics] Where did the work done by smaller force go
energy-conservationhomework-and-exercisesspringwork
Best Answer
Actually, this is incorrect. If we don't want to be adding kinetic energy and worrying about that, then the total force from the people must at all times equal the force from the spring. This means that the force from the second person is only 30 N at full extension and is less than 30 N before then. Here is a plot of the forces vs displacement. The blue line is the total force from both people, the yellow line is the force from the first person, and the green line is the force from the second person.
In this plot the work is the area under the curve. It is immediately obvious that the area under the blue curve is the sum of the areas under the curve for the yellow and for the green person.
What is not so obvious is that the work done by the first person is actually more than the work done by the second. The area under the green curve is 3.0 J, not 6.0 J. The area under the yellow curve is 3.75 J, with 0.75 J under the initial triangular portion from 0 to 0.1 m and the remaining 3.0 J in the flat section.
So, indeed, the work done by the smaller force is essential for calculating the total energy that went into the spring. The calculation of 6.0 J from the second person is simply incorrect. It is incorrect by a factor of 2 which basically is the difference between incorrectly assuming that the 30 N force was applied throughout the movement instead of correctly recognizing that it is only 30 N at full extension. Of course, alternatively you could consider the change in KE that would result from a constant 30 N force, as described by @Aaron Stevens in his answer.