The discussion revolves around calculating the energy of a spring using physics principles, specifically focusing on the work-energy theorem and the relationship between force, distance, and potential energy in springs. Participants are exploring concepts related to spring constants, potential energy, and kinetic energy in the context of a problem involving a spring system.
Some participants are attempting to clarify their understanding of the problem and the underlying physics principles. There is a mix of confusion and attempts to reason through the problem, with various interpretations being explored. Guidance has been offered regarding the relationship between force and energy, as well as the significance of the area under the force vs. distance graph.
Participants express confusion about specific parts of the problem and the calculations involved. There are references to the need for calculus in understanding the integration of variable forces and the area under the curve. Some participants question the validity of using graphical methods versus algebraic methods for solving the problem.
The first part of the graph is a graph of force versus distance. What is the slope of that portion of the graph, and how do you think that is related to the spring constant of the inner spring?muna580 said:F=kx Force for Spring
K=1/2kx^2 Kenetic Engery for spring
From the work done by the force that stopped the mass. C) is the work done while the mass moves from B to C. D) is the total work done.muna580 said:I have the solutions to this problem, but there are 3 parts in the solutions which I don't understad.
I don't understand why the answer to part C is 2J.
In part D, they say,
K(inital) = 1/2mv_o^2.
3J = (1/2)(5)v_o^2.
How did they get 3J as the inital kenetig enegery?
The reason a spring has potential energy 1/2kx^2 is because that is the amount of work it takes to stretch or compress a spring. Work is the force times the distance over which the force is applied. When the force is variable, work is found by integrating the vaiable force over the distance. If you have not had calculus, the integral is equal to the area under a force vs distance curve. The work done on the spring while the mass moves fro A to B is the area of the triangle, and from B to C it is the area of the trapezoid.muna580 said:Huh? I don't get it. I am so confused.
muna580 said:One more thing, why is ΔK = ΔU?
See the annotations in blue in the quote area.muna580 said:I am curretnly taking calculus. I am still a little confused, but check this out, so far, I understand the follwoing.
My Understing.
a) What is the k? F = kx. By looking at the graphy, after traveling .10m, the force is 20N.
F = kx
20 =.1k
k = 200
This is good, but don't forget the units
b) Since the spring is losing enegery,...
The spring is gaining energy. It is doing negative work on the mass, and the mass is doing positive work on the spring. The mass is losing kinetic energy as the spring gains potential energy.
(1/2)kx^2 = Energy
(1/2)(200)(.1^2) = 1J.
Yes. And note that this is the area under the triangle. Anytime you have a graph of force vs distance in the direction of the force you can find the work by finding the area.
c) I STILL DON"T UNDERSTAND HOW TO DO IT.
The work is the area of the trapezoid between points B and C. It is also the sum of the energies stored in the two springs. The order of the questions tells me they wanted you to do this part using the area, but you could also find the spring constant for the second spring and use the compression of the first spring from A to C, plus the compression of the second spring from B to C to find the total work (total energy stored in the springs). Then find the work between B and C, by subtracting the work done between A and B. It is much easier to just find the area under the curve.
d) I STILL DON"T UNDERSTAND HOW TO DO IT.
The total work done on the springs is the total area under the curve from A to C. This is the potential energy stored in the springs when the mass comes to rest. It is also the total kinetic energy the mass had before it started slowing down. From this energy and the kinetic energy equation you can find the initial velocity.
e) I STILL DON"T UNDERSTAND HOW TO DO IT.
The steeper slope is the combined spring constant of the two springs. You can extend the first line out to x = .15 (or use the now known spring constant of the inner spring) to see how much force is being applied by the inner spring compressed .15m. The remaining force is compressing the outer spring from B to C, a distance of .05m. If you work this out you will see that the difference between the two slopes is the spring constant of the second spring.
One more thing, why is ΔK = ΔU?
Because no work is done by any disipating force like friction. The mass does work on the springs and gives them energy. The springs do work against the mass (force in opposite direction to the motion) and take energy from it.
OlderDan said:See the annotations in blue in the quote area.
It is not cheating at all. The formula for the energy stored in a spring is a consequence of the fact that the integral of Fdx = kxdx is 1/2kx^2. We get the that formula because 1/2kx^2 is the area under the curve.muna580 said:Wow, that was awosme, I understand everybthing. BUT, big BUT, huge BUT. lol
Umm, I understood what you said for part C, but I think finding the area under the graph is cheating. :). I mean, I like to learn things the real mathemcial way. I don't even know why we are using the area under the curve so...
Ummm, can you like write out the equation/solve part C, wihtout doing the area under the curve?
Also, I know finding the area understand the curve is taking the intergral, which is the same as the anti-dervertive, but why are we taking the area under the graphy of a F vs distance? What does it give us? How why?