Block on a Spring, Nonconstant Net Force

[grandpa2390]

Homework Statement

A spring has a relaxed length of .2m and its spring stiffness is 8 N/m. You glue a .06 kg block to the top of the spring, and push the block down, compressing the spring until its total length is .1 m. You make sure the block is at rest and then quickly move your hand away. the block begins to move upward, because the upward force on the block by the spring is greater than the downward force on the block by the Earth. Make a graph of y vs time for the block during a .3 s interval after you release the block.

Relevant Equations

F{spring} = -K*S*L
F = ma
y = y_o+v_o*t+.5*a*t^2
p_f = p_i + F_net*t

Just to be clear, this isn't a homework problem. it is an example problem found on page 68 of the text "Matter and Interactions" 4th edition. The solution is given in the book, but I'm having difficulty following their reasoning.

according to the book the net force is not constant, therefore we can only get estimates by calculating iteratively.
This makes sense. gravity is constant -9.81 m/s2, but as the spring expands, the amount of force it will exert on the block becomes smaller until it reverses direction. and becomes larger until it overcomes the momentum of the block and then repeat until it overcomes gravity.

The problem is that the acceleration is varying.

So I thought, I'll assume over such a short period of time, a_avg = a_initial. let me find the net Force, use that to find the initial acceleration, then find the position after .1s. repeat.

The book solves this by assuming that over a short period of time, v_avg = v_final. so finding the net force, use the initial momentum and that to find final momentum. find the final velocity from that. Because they assume that v_final is the average velocity, they plug that into the kinematic equation y_i + v_final*t.

their final position after one iteration is y=.135 m. (.1 + .035)
My final position after one iteration is y = .118 m. (.1 + 1/2*.035)

my question is: is there an issue with how I chose to solve the problem? Am I wrong? or is my estimate a better estimate since I assumed the acceleration was constant over the iteration and factored in changing velocity, and they assumed the velocity was constant over the iteration.

I've had this issue again and again with the examples in this book. they love to assume that the final velocity is the average velocity in their estimates even though we know the acceleration. to me it seems like it makes more since to estimate the average acceleration in a problem like this than the velocity.

 

[haruspex]

I believe your method is the more accurate.

 

[grandpa2390]

I believe your method is the more accurate.

Well while I was waiting, I continued reading in the book and I may not be understanding them, but they seem to my question themselves and then respond that it is less accurate. and they show a graph that the method I used is less accurate over time. I am currently seeing if my python skills will allow me to test that.

Am I understanding this correctly. my method with the acceleration ought to be equivalent to calculating (Vf-Vi) / 2, right?

edit: I don't know. I can't get my program to plot properly.

 

[Delta2]

The exact equation we get if we solve the differential equation is $x=0.026425\cos(11.547t)$. You can compare this by the solutions of your book and your own to see who is closest. But you got to tell me how exactly you measure y (you measure it from the spring natural length, so at t=0 it is y=0.1m for example, or you measure it from the spring maximum compression or how ?) so I can modify the above equation. The above equation x=0 corresponds to compression 0.073575m from the point of natural length.

 

[haruspex]

method with the acceleration ought to be equivalent to calculating (Vf-Vi) / 2, right?

If you mean (Vf+Vi) / 2, yes.

I put the two methods and a cos function into a spreadsheet.
After a quarter of an oscillation, approaching the equilibrium point, the cos function has a displacement 0.011; your method has -0.004; the book has -0.019. So at that stage you have the more accurate displacement.
But tracking the energy, the book method lost a bit of energy to begin with, then pretty much got it all back by the quarter cycle point. Your method had smaller changes, but all in the positive direction.
So although your method is more accurate over shortish stretches, the errors accumulate.

I have a suspicion that motions other than SHM could work better with your method, but I need to do more analysis to figure out what is happening.

 

[haruspex]

more analysis

Suppose the motion is $y=\cos(\omega t)$. We can consider how two simulation algorithms update y and velocity over a time step $\delta t$ and compare the resulting error accumulated in the total energy.
To simplify matters, I only compared using the prior velocity to update the displacement with using the posterior velocity. Using the average velocity presumably gives a result between the two.
Using the prior velocity gives $\delta E=\frac 12\omega^4\delta t^2$.
Using the posterior velocity gives $\delta E=-\frac 12\omega^4\delta t^2\cos(2\omega t)$, plus terms in higher powers of $\delta t$.
Since these are equal and opposite at first, using the average velocity starts out looking best.
But note what happens summing over a cycle. The error in using the prior velocity is always positive, so accumulates without limit, whereas the error in using the posterior cancels. Remarkably, even the next term, $\frac 14\omega^5\sin(2\omega t)\delta t^3$, cancels, leaving only $\frac 12\omega^6\cos^2(\omega t)\delta t^4$ to accumulate.

 

[PeroK]

Just to be clear, this isn't a homework problem. it is an example problem found on page 68 of the text "Matter and Interactions" 4th edition. The solution is given in the book, but I'm having difficulty following their reasoning.

according to the book the net force is not constant, therefore we can only get estimates by calculating iteratively.

I've had this issue again and again with the examples in this book. they love to assume that the final velocity is the average velocity in their estimates even though we know the acceleration. to me it seems like it makes more since to estimate the average acceleration in a problem like this than the velocity.

It makes even more sense to use calculus. The book's approach seems at.odds mathematically with.taking the limit of Newton's second law - which treats force and acceleration as continuous functions.