# Hooke's and Newton's law to find Second order ODE

• Remixex
In summary, Homework Equations: The attempt at a solution was easy, but the equation was confusing and the graph was too.
Remixex

## Homework Statement

A weight of 8 pounds extends a spring 2 feet. It's assumed that the damping force that acts on the system is equal (number-wise) to alpha times the speed of the weight.
Determine the value of alpha > zero so x(t) is critically damped.
Determine x(t) if the weight is liberated from it's equilibrium position with a descending speed of 3 feet per second
Graph x(t)

## Homework Equations

I know i need to use Hooke's and then Newton's law

## The Attempt at a Solution

I'm fully aware of how to solve a second order differential equation.
I also know that, in the end, i have to achieve
x´´(t) + 4ax´(t)+16x(t)=0 ///
x(0)=0
x´(0)=3

Since that second order ODE has no g(x) i just use the homogeneous formula with the initial conditions to create the function i need.
I also know alpha needs to be 2 so this is a critically damped system because, here, the frequency squared is 16 and 2 times the number alongside x´(t) needs to be equal to the frequency (2a=w) => a=2

I found this problem in my DE course at university and i couldn't solve it because i couldn't even get to the problem, i am now trying to find sense to it but i need some directions

Remixex said:
I know i need to use Hooke's and then Newton's law
This is a good start, what do you get when you try doing so?

Also, I suggest using symbols (such as m for the mass and L for the extension of the spring) rather than insering the numbers from the beginning. This will make it much easier to see if you are doing things correctly or not.

Ok so with Hooke's law we have that the force which the string is pulled it's proportional to the distance the string expanded
It seems my teacher is a bit weak on Physics notation but i think that by 8 pounds he actually meant the weight force
So we get 8=2K therefore K=4
Then on the solution he just establishes 8=Mx32 with M being mass, this is where i get lost, i have no idea how to get there
He then applies Newton's second law of motion to establish 1/4x´´(t) = -4x(t) - ax´(t)

I know that the "-ax´(t)" comes from "the damping force is equal to alpha times the speed of the weight"
the "-4x(t)" comes from Hooke's law, , minus K times the displacement
and "1/4 x´´(t)" comes from mass times the acceleration
I just need to know how did he calculate mass, where did 8=Mx32 come from

Remixex said:
Then on the solution he just establishes 8=Mx32 with M being mass, this is where i get lost
This is where everyone gets lost and the reason why you should either use variables to represent your given quantities or change to SI units. Just putting numbers will only act to confuse you.

The 32 is a conversion factor between lbf and lbm (awful system of units that one).

Remixex
My goodness...i haven't had a physics course in 1 year and the teacher expects me to remember a conversion factor for a really important DE test.
Thank you very much
The equation was so easy to solve and the graph too, but i couldn't even begin the problem :,c

Orodruin said:
The 32 is a conversion factor between lbf and lbm (awful system of units that one).

No, there is nothing at all wrong with the units system if it is used correctly. The fact that it is often screwed up is irrelevant; SI is subject to the same screw up.

The 32 is a (very, very rough) approximation to the acceleration of gravity. The real confusion lies in talking about lbf and lbm. The pound unit properly applies to force and only to force; the idea of a pound mass is a fiction invented for the convenience of people who catch things in buckets and want to weigh them to find out how much they have caught. In SI countries, we see the use of kg and kgf, where the kgf the gravitational force on a 1 kg mass.

There is little or nothing at all superior about the SI system. It solves no problems any better than the US Customary system does. Either can, and often are, used inconsistently, but that is not a fault of the system.

When I was a young engineer, back in the early 1960s, I though the SI system was just wonderful, the solution to all difficulties, and was inevitable. Now that I am an old man, I have learned through the years that the choice of unit systems makes no difference for mechanics problems; either works just fine if it is properly used. Neither is superior. (Sure, I know all about the powers of 10 and prefixes, but does a terameter really mean anything to you until we convert it to standard scientific notation? It certainly does not mean much to me, other than maybe a device for measuring teras. Now, try dividing you terameter by 3 or 6.)

Quite knocking the US Customary system. It works fine, and at present, it looks like it will continue to be in use for a very long time.

OldEngr63 said:
No, there is nothing at all wrong with the units system if it is used correctly.

But this exactly the point. The imperial unit system is full of conversion factors between different units for the same physical quantity which are not adapted to the fact that we are generally using base 10. In addition, the fact that both lbf and lbm go as "pounds" is utterly confusing. There is a reason the SI system is more widespread in science and engineering.

Of course you can use any system as long as you use it right, it is the "using it right" which is the problem.

Then again, as a particle physicist, I usually consider unit systems where c and hbar are not one as a bit weird.
OldEngr63 said:
In SI countries, we see the use of kg and kgf, where the kgf the gravitational force on a 1 kg mass.
I disagree with this. I have not seen kgf used to any extent. The SI unit of force is Newton.

Orodruin said:
The imperial unit system is full of conversion factors between different units for the same physical quantity
But that is exactly the point! Force and mass are different and nobody who understand the difference uses both lbm and lbf. Maybe you missed my point about "people who catch things in buckets" but I was referring to thermodynamicists and fluid mechanicians who are the only people I know about who use both lbm and lbf. The tend to want to know the mass of a fluid coming out of a pipe, so they catch the amount delivered in 10 sec and then weigh it. This is how they get into the lbm mess!

Orodruin said:
The imperial unit system
The British Empire abandoned inches, feet, pounds, etc. quite a few years ago, so it makes no sense at all for folk to continue to speak of the Imperial Unit system. The use of such units today is limited to the USA, where it is called the US Customary Unit system.

Orodruin said:
There is a reason the SI system is more widespread in science and engineering.
No, not really. In science, probably so, but in American engineering, not so at all. It simply has not happened. The textbook publishers love SI units and most texts today are straight SI. Practicing engineers still use US Customary units almost exclusively. The simple fact is, SI has found nothing to really recommend itself to engineers, and they will not willingly change without a reason.

With reference to kgf, you said
Orodruin said:
I disagree with this. I have not seen kgf used to any extent.
You obviously have not read any of the older European engineering literature. Stress in kgf/mm^2 is very common (not very good SI, but still common). They did not know about the Pascal (a truly worthless unit; it is far, far too small for engineering purposes).

And that brings me to another point about SI in general. The units are the wrong sizes. A meter is no more useful than a yard, and what technical use (football excluded) have you found for a yard? You don't measure the area of a space in sq yards (oh, I forgot; cloth is still sold by the sq yard, I think). If you want to measure a large distance, a mile is better than a kilometer - no prefix needed. If you want to measure the thickness of a piece of sheet metal, a meter value is worthless, so you have to switch to a mm - back to prefixes. Inches work better. For stresses in Pa, the number are often outlandishly large. This has brought back the semi-SI unit the bar = 10^5 Pa that is just a little under 14.7 psi. It goes on and on. For mechanics, SI units are not one lick superior to US Customary units. For electromagnetic and other similar things, SI is the only way to go, but not for mechanics.

Orodruin said:
it is the "using it right" which is the problem.
It is really just like any other tool. If taught/learned wrong, as it usually is by physicists, then trouble will be the result. I've been using both unit systems, back and forth, interchangeably for 50+ years without a problem. It only requires a clear grasp of the difference between force and mass. Without that, you really can't do mechanics.

It has always appeared to me that particle physicists are a bit weird; they spend their entire lives thinking about and chasing things that they have never seen and likely never will. By all means, they should use on SI and insist on it for their transactions at the grocery store, the gas station, etc -- i.e. in the real world where most people live.

The most striking thing is that i live in Chile, we use the MKS system for everything here.
It seems my teacher just translated a problem from a book and expected us to know how pounds and acceleration works with them, which is quite controversial in my opinion.
In fact i don't even know how much a foot is exactly

OldEngr63 said:
You obviously have not read any of the older European engineering literature.

Your sentence was posed using present tense. I see no reason to discuss things that are obsolete.

OldEngr63 said:
If you want to measure a large distance, a mile is better than a kilometer - no prefix needed.
No, this is not objectively true. I would consider using a prefix superior to defining a new length unit. Most of all, based on our number system, the conversion between units with different prefix is trivial.
OldEngr63 said:
By all means, they should use on SI and insist on it for their transactions at the grocery store, the gas station, etc -- i.e. in the real world where most people live.
Which is working perfectly, thank you very much. In the real world, where most people live (you know, outside the US), this is done to a large extent.
OldEngr63 said:
For stresses in Pa, the number are often outlandishly large. This has brought back the semi-SI unit the bar = 10^5 Pa that is just a little under 14.7 psi. It goes on and on.
You are bringing up a non issue (thanks to the existence of prefixes!). MPa or kPa are perfectly fine SI units. Named units such as bar and liters are introduced because they are handy, but they still conform to the use of units which differ by multiples of 10 and therefore allow for trivial conversion.

OldEngr63 said:
It has always appeared to me that particle physicists are a bit weird; they spend their entire lives thinking about and chasing things that they have never seen and likely never will.
http://arxiv.org/abs/1207.7214
You are also perfectly welcome to stop using derivatives of particle physics research, you know, like the www, computers, etc. The point of research is that you do not know what is going to happen.

Regardless, this discussion is going off topic and the OP's question has been resolved. Thread closed.

Last edited:
e.bar.goum

## 1. What is Hooke's law?

Hooke's law is a principle in physics that states that the force needed to extend or compress a spring by some distance is directly proportional to that distance. In other words, the more a spring is stretched or compressed, the greater the force applied to it.

## 2. How is Hooke's law related to Second order ODE?

Hooke's law can be used to model the behavior of a spring-mass system, which is described by a second order ordinary differential equation (ODE). The equation is derived from Newton's second law of motion and can be used to find the position and velocity of the mass at any given time.

## 3. What is Newton's law of motion?

Newton's law of motion is a set of three physical laws that describe the relationship between the forces acting on an object and its motion. The first law states that an object at rest stays at rest and an object in motion stays in motion with constant velocity unless acted upon by an external force. The second law relates the mass and acceleration of an object, and the third law states that for every action, there is an equal and opposite reaction.

## 4. How is Newton's law of motion used to find Second order ODE?

Newton's second law, which states that force is equal to mass times acceleration, can be used to derive the second order ODE that describes the motion of a spring-mass system under the influence of an external force. This equation can then be solved to find the position and velocity of the mass at any given time.

## 5. What are some real-life examples of Hooke's and Newton's law?

Examples of Hooke's law in action include a car's suspension system, a diving board, and a rubber band. Newton's laws can be observed in many everyday situations, such as a ball rolling down a hill, a rocket launching into space, and a person pushing a shopping cart.

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