Understanding the meaning of maximal contraction values

[assaftolko]

suppose I have a horizontal problem where I have a spring attached to a wall, a body that moves toward it in some inital speed and the surface isn't smooth so I have friction along the surface towards the spring and also while the body contracts the spring. I'm asked what's the maximal contraction? I get 2 values for Xmax: one positive and the other negetive (also different in absolute value). One of them is 0.149 m and the other is -0.208. How can I know which one represents maximal contraction and which one represents maximal strechning?

this is a general question - never mind the specific values of k, mue(k), distance from spring at which body starts from with vo etc...

 

[gneill]

Whether to select a positive or negative result will depend upon your choice of axis origin and direction. You should carefully define your axes before you start writing equations, and be consistent in their application to the equations.

 

[assaftolko]

Whether to select a positive or negative result will depend upon your choice of axis origin and direction. You should carefully define your axes before you start writing equations, and be consistent in their application to the equations.

the axis isn't my choice - I uploaded the question - I still don't get what difference does it make... i didn't need to consider the axis direction all along the problem: not for calculating fk, not for calculating the work fk does (which is -mue(k)mg(1+Xmax) where 1 is the distance from where m has v0 of 3 m/s (starting point) to the edge of the spring) so as far as I see it, until this point I could have done this prob without even drawing the positive direction of the x axis...

 

[gneill]

Yes, but when you write your energy expressions you are making an assumption about what a positive value of displacement represents. Given your choice, does a negative value for displacement make sense?

 

[assaftolko]

Honestly i don't know... I would think that maybe if the positive direction of x-axis is to the right then for contraction (which is to the left) i would in fact want a negative value, but if i would have chosen the positive axis to the other side i still would have goten the same values for Xmax with the same signs, because the x-axis direction didnt affect my calculations at all... I am confused

 

[gneill]

Okay, here's a suggestion for a way to investigate the problem that may help. Write an expression for the KE remaining in the body with respect to x from the moment it first touches the spring. You can reset x=0 for that point. Graph the result and observe the form of the curve; there should be two locations where the curve crosses the x-axis (KE = 0). Does your choice of x direction influence the root that you should select? Which root occurs along a trajectory moving forward in time?

 

[assaftolko]

Okay, here's a suggestion for a way to investigate the problem that may help. Write an expression for the KE remaining in the body with respect to x from the moment it first touches the spring. You can reset x=0 for that point. Graph the result and observe the form of the curve; there should be two locations where the curve crosses the x-axis (KE = 0). Does your choice of x direction influence the root that you should select? Which root occurs along a trajectory moving forward in time?

I'm sorry but I don't quite understand how to do this:
Write an expression for the KE remaining in the body with respect to x from the moment it first touches the spring

 

[gneill]

What's the KE in body when it just reaches the spring? It begins its life with KE corresponding to its initial speed and mass, then proceeds to lose KE to friction until it reaches the spring. So how much KE does it have left when it reaches the spring?

Now, write an expression for the KE from that point forward as the spring compresses (trading some KE for spring PE) and also continues to lose KE to friction. Plot that expression with x as the independent variable. That is, KE(x).

 

[assaftolko]

What's the KE in body when it just reaches the spring? It begins its life with KE corresponding to its initial speed and mass, then proceeds to lose KE to friction until it reaches the spring. So how much KE does it have left when it reaches the spring?

Now, write an expression for the KE from that point forward as the spring compresses (trading some KE for spring PE) and also continues to lose KE to friction. Plot that expression with x as the independent variable. That is, KE(x).

as the body reaches the spring it has speed of 1.766 m/s, so its KE at that point is 1.56 J. the KE from here on is equal to: Wfk-Eu = -fkX-0.5kX^2 and fk=mue(k)*mg=0.3*1*9.8=2.94N, so:

KE(x)= -2.94x-50x^2+1.56 is this correct?

 

[gneill]

as the body reaches the spring it has speed of 1.766 m/s, so its KE at that point is 1.56 J. the KE from here on is equal to: Wfk-Eu = -fkX-0.5kX^2 and fk=mue(k)*mg=0.3*1*9.8=2.94N, so:

KE(x)= -2.94x-50x^2 is this correct?

Almost. Don't forget to include the "initial" KE of 1.56J.

If you plot this expression w.r.t. x, you should note that there are two points where KE is zero and that these points correspond to the two values that you solved for earlier. But only one of them occurs "forward in time" from the moment of contact of the body with the spring. Which one is forward in time depends upon your assumption of direction of the x-axis, and what it means for x to increase (or decrease). This assumption was made, regardless if consciously or unconsciously, when you wrote the conservation of energy equation.

Your equation only works for x going in one direction, since it was written to have frictional energy loss as x increases. If you were to let x run backwards (go negative) in this equation, the system would GAIN energy from friction, which is clearly a nonphysical result.

 

[assaftolko]

Ok so I get a parabula with roots at -0.208 m and 0.149 m just as I got from my initial calculation - how can I know from this which one corresponds with the forward movement of time?

 

[assaftolko]

Almost. Don't forget to include the "initial" KE of 1.56J.

If you plot this expression w.r.t. x, you should note that there are two points where KE is zero and that these points correspond to the two values that you solved for earlier. But only one of them occurs "forward in time" from the moment of contact of the body with the spring. Which one is forward in time depends upon your assumption of direction of the x-axis, and what it means for x to increase (or decrease). This assumption was made, regardless if consciously or unconsciously, when you wrote the conservation of energy equation.

Your equation only works for x going in one direction, since it was written to have frictional energy loss as x increases. If you were to let x run backwards (go negative) in this equation, the system would GAIN energy from friction, which is clearly a nonphysical result.

Well this is a bit difficult for me to understand - i think i see your main point but the "Eureka!" isn't quite there yet... how would the problem cahnge if the positive direction of the x-axis was chosen to be to the left insted of to the right?

 

[gneill]

Well this is a bit difficult for me to understand - i think i see your main point but the "Eureka!" isn't quite there yet... how would the problem cahnge if the positive direction of the x-axis was chosen to be to the left insted of to the right?

Well actually, that's just what you did when you wrote your equations! You assumed increasing x going to the left. Otherwise your friction term would have to have the opposite sign, and you would expect the result to be a negative value for x. This is what I mean about assumptions being made, whether or not you make them consciously.

Note that it's okay to write equations without slavishly adhering to an initial choice of coordinate axes provided that you understand the relationships between the quantities that you're using and the way they each behaves with the "impromptu" choice of axes. You will gain comfort with experience.

Now, suppose that you consider for a moment what it would mean if you chose the negative solution. Would that correspond to a compression or an extension of the spring? How could the spring become extended before the body even reaches it?

 

[assaftolko]

Well actually, that's just what you did when you wrote your equations! You assumed increasing x going to the left. Otherwise your friction term would have to have the opposite sign, and you would expect the result to be a negative value for x. This is what I mean about assumptions being made, whether or not you make them consciously.

Note that it's okay to write equations without slavishly adhering to an initial choice of coordinate axes provided that you understand the relationships between the quantities that you're using and the way they each behaves with the "impromptu" choice of axes. You will gain comfort with experience.

Now, suppose that you consider for a moment what it would mean if you chose the negative solution. Would that correspond to a compression or an extension of the spring? How could the spring become extended before the body even reaches it?

Of course it can't, but the mathmatical solution doesn't "know" this - I still don't quite understand then what the negative solution represents... and if I was told that the x-axis is positive to the right - would my solution as i did it here be correct? I understand the values are correct but i don't understand if my way corresponds with the initial choice of coordinates as it was given to me and if indeed i should have solved it in another way - what should have I done differently?

 

[gneill]

Of course it can't, but the mathmatical solution doesn't "know" this - I still don't quite understand then what the negative solution represents... and if I was told that the x-axis is positive to the right - would my solution as i did it here be correct? I understand the values are correct but i don't understand if my way corresponds with the initial choice of coordinates as it was given to me and if indeed i should have solved it in another way - what should have I done differently?

Your solution would not have been correct if you were told explicitly to assume the x-axis is to the right. After all, your initial velocity would then have to be negative, and your friction term would have to account for friction loss increasing as x decreases. The math would have been complicated by an imposed choice.

You simplified your life by making an intelligent choice of axis orientation and origin, which made the math simpler. But it's up to you to understand what the implications are of the choices you make and to interpret the results in terms of "physically possible".

 

[assaftolko]

Your solution would not have been correct if you were told explicitly to assume the x-axis is to the right. After all, your initial velocity would then have to be negative, and your friction term would have to account for friction loss increasing as x decreases. The math would have been complicated by an imposed choice.

You simplified your life by making an intelligent choice of axis orientation and origin, which made the math simpler. But it's up to you to understand what the implications are of the choices you make and to interpret the results in terms of "physically possible".

But as you see from what i uploaded they chose x to be positive to the right... Can you please solve the problem for the positive to right x axis? I think it all would be clearer to me if ill see the other solution

 

[gneill]

But as you see from what i uploaded they chose x to be positive to the right... Can you please solve the problem for the positive to right x axis? I think it all would be clearer to me if ill see the other solution

Unfortunately, we're not permitted to provide complete solutions to homework problems. We can only help you to reach a solution with guidance, hints, references, explanation of concepts, and so on.

If you want to pursue the "enforced choice of axes" solution, I can help you. The first thing to do would be to make explicit your choice of axis by stating the direction and origin, then locate the important positions along the axis (like the initial position of the mass, and the position of the relaxed spring. A clever choice of origin will help.

Regarding the "specious" negative solution to your original method, note that in reality the physics of the problem changes when the mass reaches the spring; it is only then that the potential energy of the spring comes into play. The equation you wrote made the assumption that the mass had reached the spring, having shed some KE to friction on the way. But as you put it, the math doesn't know this. It could equally well represent a situation where the mass is moving in the other direction and remains attached to the spring, thus stretching it while gaining KE from friction! This is not a physically viable solution, doesn't represent what you intended, and so should be discarded. But it's up to you to recognize this, since you're writing and interpreting the equations.

 

[assaftolko]

but why does it "gain" energy when it streches the spring? friction will act to the left, which is the negative direction with respect to the coordinate system given to me...

 

[gneill]

but why does it "gain" energy when it streches the spring? friction will act to the left, which is the negative direction with respect to the coordinate system given to me...

It's just a possible interpretation of the equation as written, even if you didn't originally intend it that way. It's up to you to assign the sign to the changes in KE and PE, since sign information is "lost" in the squaring operation of velocity and position in their formulas. And frictional should always cause a loss of energy going forward in time, but the simple expression -f*d for the energy lost is symmetrical with respect to time and distance and would yield a gain in energy for negative values of d. This is okay as long as you understand your intentions with the equations (like if you were projecting backwards in time to find a given situation in the past).

It is up to you to write and interpret the equations in such a way that they correspond to your understanding of the physics taking place. When equations are used outside their region of applicability, all bets are off.

 

[assaftolko]

It's just a possible interpretation of the equation as written, even if you didn't originally intend it that way. It's up to you to assign the sign to the changes in KE and PE, since sign information is "lost" in the squaring operation of velocity and position in their formulas. And frictional should always cause a loss of energy going forward in time, but the simple expression -f*d for the energy lost is symmetrical with respect to time and distance and would yield a gain in energy for negative values of d. This is okay as long as you understand your intentions with the equations (like if you were projecting backwards in time to find a given situation in the past).

It is up to you to write and interpret the equations in such a way that they correspond to your understanding of the physics taking place. When equations are used outside their region of applicability, all bets are off.

Ok thanks a lot!