# Homework Help: Understanding why the initial height of one block is equal t

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1. Jan 11, 2016

### RoboNerd

1. The problem statement, all variables and given/known data
A 20.0-kg block is connected to a 30.0-kg block by a string that passes over a light frictionless pulley. The 30.0-kg block is connected to a spring that has negligible mass and a force constant of 250 N/m. The spring is unstretched when the system is as shown in the figure, and the incline is frictionless. The 20.0-kg block is pulled20.0 cm down the incline (so that the 30.0-kg block is 40.0 cm above the floor) and released from rest. Find the speed of each block when the 30.0-kg block is 20.0 cm above the floor (that is, when the spring is unstretched.

The problem is on page 62 of the document on the attached website along with the diagram.

http://www.santarosa.edu/~lwillia2/40/40ch1011_f14.pdf

2. Relevant equations:
Conservation of mechanical energy. Mechanical Energy initial = Mechanical Energy final

3. The attempt at a solution

I prove that I attempted to solve the problem by attaching a picture below of my unfinished sheet.
So my high school teacher assigned the problem above. I just took the link from a college website because it has the problem with diagram and the explanation. I also have the following photographs from the student solutions manual that the book uses, which are attached below.

I understand how conservation of mechanical energy works in this case, but I am struggling with another aspect of the problem.
So I am struggling to understand one critical aspect of their explanation.... why does the height of the 20.0 kg block when it is pulled back equal the height of the 30.0 kg block when the spring becomes unstreched?

I would have thought that because there is a 40 degree angle that things would have been different....

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2. Jan 11, 2016

### Simon Bridge

Clearly the height of the block depends on the length of the string connecting them together (and, yes, the angle of the ramp).
Do they tell you the length of the string? Does the length matter for the thing you have to find out?

3. Jan 11, 2016

### RoboNerd

Nothing was said about the string in any way shape or form.

4. Jan 11, 2016

When they say "The 20.0-kg block is pulled 20.0 cm down the incline" they mean the vertical height, not the actual length of the ramp. Indeed the length down the ramp is:

$L = \frac{h}{sinθ}$

5. Jan 11, 2016

### RoboNerd

So how does this prove that the initial position of the 20.0 kg block is equal to the final position of the 30.0 kg block?

6. Jan 11, 2016

### WrongMan

The block on the incline (lets call it block 1) is pulled down 20 Cm.. which (coincidentally) is the same height that the other block is when the system is at rest.
there is no logical explanation as to why the block1 is pulled down 20 cm, its just the way the exercise is set, if the block1 were to be pulled down 50 cm the rest position for the system (or unstreached position for the spring) would still be with block2 at 20cm

now, if you meant, why is it that when you pull block1 20 cm, the other block also rises 20 cm, then thats just logic, 20cm is 20cm in whatever direction you mesure it...the distance block1 travels is exacty the same distance block2 travels, just in diferent direction

7. Jan 11, 2016

### RoboNerd

No, I am not asking that. I am told that the vertical height of the block that is pulled 20 cm at the point where it is pulled away is equal to the vertical height of the equilibrium position of the 30 kg block at equilibrium....

so I have the following... as shown in my photograph.

Why is this the case?

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8. Jan 11, 2016

It's just a coincidence, it just means the 20 kg block was at an initial height of 40 cm.

9. Jan 11, 2016

### RoboNerd

But they do not give me the initial height so logically, I can not even derive the fact that the 20 kg block was at an initial height of 40 cm.
And thus, I can not logically derive what the solutions establish to be fact: that the height of the 20 kg block when it is pulled back equals that of the 30 kg block at equilibrium height.

.... so how could I solve the problem without realizing (or making up) this "coincidence?"

10. Jan 11, 2016

### WrongMan

I don't see anywhere that says that the block1 when pulled 20cm along the incline ends up at an height of 20cm relative to the horizontal... even if it does, that is just a pure coincidence... it certainly doesnt say that (in relation to the horizontal) its original position had an height of 40cm... that just means his original height was 20+20sen(θ)

PS: i cant read the solutions photograph

11. Jan 11, 2016

### Simon Bridge

What's happening is prob that the thing you are trying to find does not depend on the length of the string (and thus the height of the block) ... so you are free to pick any length you like for that string.
The best thing to do is to pick a length that makes the math simpler for you.
The authors chose the length so that the height of the black on the ramp is the same as the height of the other one when the spring in uncompressed.
You want to see how that changes things - try using a different variable for the height of the block on the ramp.

12. Jan 11, 2016

You don't need this information, all you know is that the height of the 20 kg increases decreases by 20 cm and the height of the 30 kg increases by 20 cm, that is enough to solve the problem.

13. Jan 11, 2016

### RoboNerd

A friend of mine said he was able to solve the problem by adjusting various string lengths, but that seems a little unsystematic to me.... Would such an approach work for sure?

14. Jan 11, 2016

### RoboNerd

Ohh....

I am attaching a new photo of the solutions manual here. Hope this helps

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15. Jan 11, 2016

### RoboNerd

A friend of mine said that he was able to solve the problem by adjusting rope lengths, but this seems a little unsystematic.... and I am hesistant to do this... plus adding in a string length of my own choosing can mess up the problem, can it not?

16. Jan 11, 2016

### Simon Bridge

It doesn't matter what the string length is ... the only reason to fiddle with it is to make some numbers cancel out.
It is better practise to assign the string length to an unknown variable, say "s".
In general - do all the algebra in symbol form and only substitute the numbers later.

17. Jan 11, 2016

### WrongMan

Its a solutions typo, where they switched the masses, the pdf file you inserted has the correct form... it should be 20*0.2*sen(θ)*g and 30*g*0.2 not the other way around

18. Jan 11, 2016

### RoboNerd

Good idea... I definitely have to try that.

Question: If I have my string length "s"... then how would I represent the length of the string parallel to the incline and the length of the string hanging vertically?

19. Jan 11, 2016

### RoboNerd

WrongMan, what line of the solutions manual are you referring to?

20. Jan 11, 2016

### WrongMan

the part where they "particularize to the problem situation" and the part where they substitute for values.
they switched the masses. where m1 is there should be an m2, where m2 is there should be an m1.

or not im squinting to much

21. Jan 11, 2016

### RoboNerd

They define the 30.0 kg mass to be mass 1.... do you also have this in mind?

22. Jan 11, 2016

### WrongMan

forget about it, the solutions are alright.
what they do is they set an "imagined" lenght of string you dont need to know it, they just set the initial height of the 20kg mass (initial as in when it is pulled) to be the 0 of the referential, (and make it so its actual height is 20cm), so that the final potential energy of the 20kg mass is 0.2(its displacement)*sen(θ)*g and the final potential gravitational energy of the 30kg mass is 0.
the lenght of the string does not matter.

23. Jan 11, 2016

### RoboNerd

So it is a perfectly valid move to assume this.... and how would I determine in a future problem that the length of the string does not matter, for future reference?

24. Jan 11, 2016

### WrongMan

you have to conceptualize the problem...
in this case, imagine the triangle (slope) is on the floor, and the spring is underground, and its rest (unstreached) position makes it so the 30kg block is resting on the floor...
and you pull back on the 20kg block. you can make it so the original postion of the 20kg block minus its displacement makes it touch the floor, you have to understand that the postion of the 20kg block on the slope does not matter for the problem.
its just something you have to do, look at the problem form various angles until you find the best way to solve it...

and fyi the lenght of the string usually never matters in this sort of problems

25. Jan 11, 2016

### RoboNerd

Thanks so much for that idea!

I think I finally understand enough to do what the solutions says without feeling guilty about not understanding it!