# Block on a Spring, Nonconstant Net Force

• grandpa2390
In summary, the conversation discusses the use of iterative calculations to estimate the net force and resulting motion of a block on a spring. The book's approach assumes a constant average velocity, while the person suggests using the average acceleration. However, further analysis shows that the book's method may be more accurate over longer periods of time due to the accumulation of errors. The conversation also brings up the use of calculus in solving the problem.
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.

I believe your method is the more accurate.

haruspex said:
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.

#### Attachments

• WX20210824-154509@2x.png
46.1 KB · Views: 111
Last edited:
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.

grandpa2390 said:
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 said:
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.

grandpa2390 said:
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.

## 1. What is a block on a spring?

A block on a spring is a simple physics experiment that involves a block attached to a spring. The block is usually placed on a flat surface and the spring is stretched or compressed, causing the block to oscillate back and forth.

## 2. What is a nonconstant net force?

A nonconstant net force is a force that changes in magnitude and/or direction over time. In the context of a block on a spring, this means that the force acting on the block is not constant and may vary as the block moves.

## 3. How does a nonconstant net force affect the motion of the block on a spring?

A nonconstant net force can cause the block to accelerate or decelerate, depending on the direction and magnitude of the force. This can result in changes in the amplitude and frequency of the block's oscillations.

## 4. What factors can cause a nonconstant net force in a block on a spring experiment?

Some factors that can cause a nonconstant net force in a block on a spring experiment include friction, air resistance, and external forces such as gravity or a person pushing or pulling on the block.

## 5. How is a nonconstant net force calculated in a block on a spring experiment?

The nonconstant net force can be calculated by taking into account all the forces acting on the block and finding the vector sum of these forces. This can be done using Newton's second law of motion, which states that the net force on an object is equal to its mass multiplied by its acceleration.

Replies
9
Views
2K
Replies
29
Views
2K
Replies
4
Views
2K
Replies
61
Views
1K
Replies
33
Views
582
Replies
5
Views
3K
Replies
10
Views
1K
Replies
2
Views
856
Replies
7
Views
2K
Replies
6
Views
386