Force on the bottom of the vessel containing a submerged cube

[manasi bandhaokar]

Homework Statement

Conside the fig below. A tank of water (height of water column b) is kept on a electronic weighing scale . a metal cube (side a and density D) is hung from a spring balance and the spring balance is slowly lowered into the tank till the cube reaches the bottom of tank. The distance between the bottom of the tank and the bottom of the cube at any istant is h with initially h = h° .
I have attached the figure file with this question.
So basically we need to sketch a graph between the reading of spring balance v/s h and graph between electronic scale reading v/s h.

Homework Equations

Force of buoyance = wt of water displaced

The Attempt at a Solution

[/B]
The spring balance reading is mg -B (m is the mass of block. Assumed for the purpose of que. Not given. B is force of buoyancy. No problems here.
The problem is with electronic scale reading. It will be equal to the total downward force on the container bottom right? I thought that it should be w +mg -B (w is wt of water in the container).si this should be the reading of the scale. But the ans key I got States that the reading should be mg +B. How is it so?
We also have a sub que asking us to plot the graph of sum of readings of electronic scale and spring balance.which in ans is given as straight line parallel to X axis. Of course going by the ans key , the sum is2mg. But I don't understand how is the electronic scale reading mg +B. Or is the ans key wrong?

 

[Abhishek kumar]

Homework Statement

Conside the fig below. A tank of water (height of water column B) is kept on a electronic weighing scale . a metal cube (side a and density D) is hung from a spring balance and the spring balance is slowly lowered into the tank till the cube reaches the bottom of tank. The distance between the bottom of the tank and the bottom of the cube at any istant is h with initially h = h° .
I have attached the figure file with this question.
So basically we need to sketch a graph between the reading of spring balance v/s h and graph between electronic scale reading v/s h.

Homework Equations

Force of buoyance = wt of water displaced
[/B]

The Attempt at a Solution

The spring balance reading is mg -B (m is the mass of block. Assumed for the purpose of que. Not given. B is force of buoyancy. No problems here.
The problem is with electronic scale reading. It will be equal to the total downward force on the container bottom right? I thought that it should be w +mg -B (w is wt of water in the container).si this should be the reading of the scale. But the ans key I got States that the reading should be mg +B. How is it so?
We also have a sub que asking us to plot the graph of sum of readings of electronic scale and spring balance.which in ans is given as straight line parallel to X axis. Of course going by the ans key , the sum is2mg. But I don't understand how is the electronic scale reading mg +B. Or is the ans key wrong?[/B]

Can you post complete image of question?

 

[manasi bandhaokar]

Can you post complete image of question?

I know the pics messy but can't help it. I did all that during my test wherin it was asked.

 

[manasi bandhaokar]

I think I get it... Please verify if I am right
The effect of adding the block is that the level of water level rises in the tank.
Rise of water level is (a^3)/A. A is area of base of tank. Hence force on the bottom is-
[dg(b + (a^3)/A]A (d- density of water)
=dgbA + dg(a^3)
= w + B (I think the only mistake in the key is that when they say the electronic scale reading is mg + B the 'mg' is w actually . is this right?

 

[Abhishek kumar]

I think I get it... Please verify if I am right
The effect of adding the block is that the level of water level rises in the tank.
Rise of water level is (a^3)/A. A is area of base of tank. Hence force on the bottom is-
[dg(b + (a^3)/A]A (d- density of water)
=dgbA + dg(a^3)
= w + B (I think the only mistake in the key is that when they say the electronic scale reading is mg + B the 'mg' is w actually . is this right?

May be i can help you tomorrow

 

[manasi bandhaokar]

Err... Alright.

 

[haruspex]

= w + B (I think the only mistake in the key is that when they say the electronic scale reading is mg + B the 'mg' is w actually . is this right?

That's right.

 

[manasi bandhaokar]

So we count that the only effect of putting the block is rise if water level that changes the force on the base. Nothing else? No other effect?

 

[Abhishek kumar]

So we count that the only effect of putting the block is rise if water level that changes the force on the base. Nothing else? No other effect?

Weight of spring mass scale system depend on how force kx is set up in string and in electronic scale by normal reaction.have you got your sequence of ans like this A-R,B-Q,C-S,D-T?

 

[manasi bandhaokar]

Yeah! But I was getting A -Q and B- R.I can see why I was getting B wrong -I had got the force wrong then. But I had taken spring balance reading as mg -B . still I got it wrong. I will try it once more.

 

[Abhishek kumar]

Weight of spring mass scale system depend on how force kx is set up in string and in electronic scale by normal reaction.have you got your sequence of ans like this A-R,B-Q,C-S,D-T?

For A kx=mg-B when cube is just above the water level there is no buoyancy force as it dipped slightly buoyancy force start acting and gradually increase until cube is fully dipped and after that it remains constant now analyse graph for this you got R as right answer.

 

[manasi bandhaokar]

OK thank you very much to everyone.

 

[manasi bandhaokar]

So kx = mg - B . as the block is lowered the the reading is constant from h° to b. Then B starts increasing and the reading decreases too, as h decreases from b to b-a. After that the B is constant till h becomes 0. If you put all this together , you get Q as ans not R , where clearly as h is decreasing , reading is increasing .

 

[Abhishek kumar]

So kx = mg - B . as the block is lowered the the reading is constant from h° to b. Then B starts increasing and the reading decreases too, as h decreases from b to b-a. After that the B is constant till h becomes 0. If you put all this together , you get Q as ans not R , where clearly as h is decreasing , reading is increasing .

If you take initial position as reference point then h increase as you go down

 

[manasi bandhaokar]

But it's given that h is distance between bottom of vessel and bottom of cube which is decreasing!

 

[Abhishek kumar]

But it's given that h is distance between bottom of vessel and bottom of cube which is decreasing!

I think its matter of refrence point you choose bottom i choose height h from bottom. I choose because cube is moving gradually upto h distance. In these cases graph is just mirror image to each other may be i misinterpreted the question. BTW what is correct answer?

 

[manasi bandhaokar]

The correct matching(ans) is the one you got. But haven't they clearly specified h. And the way they have defined it means it's decreasing as block lowers

 

[Abhishek kumar]

The correct matching(ans) is the one you got. But haven't they clearly specified h. And the way they have defined it means it's decreasing as block lowers

Look if you see the motion of block h is increasing as cube is at intial positions h=0 and as it reaches bottom h=ho and its easy to start with top most point

 

[manasi bandhaokar]

Yeah I get that. But it's said that the INITIAL position is h = h° . when we consider the ref point as you considered , the initial position is h= 0. Probably the paper setters messed up with the origin so that the question says something and options say the opposite , that's why my ans doesn't match.

 

[haruspex]

Yeah I get that. But it's said that the INITIAL position is h = h° . when we consider the ref point as you considered , the initial position is h= 0. Probably the paper setters messed up with the origin so that the question says something and options say the opposite , that's why my ans doesn't match.

I am not clear on what you think the answer is and what the official answer is.
The time order of events does not matter. The graphs, unless otherwise noted, should be asumed to be h=0 on the left. On that basis, the correct answer should be AS, BR, etc.

 

[manasi bandhaokar]

The correct ans is A-R, B-Q , C-S , D -T. And I am getting A -Q , B-R.

 

[haruspex]

The correct ans is A-R, B-Q , C-S , D -T. And I am getting A -Q , B-R.

Sorry, you are right. It does look as though the official answers are reversed. And it should be C-P.

 

[manasi bandhaokar]

Well but , as u said I was right about the only effect of addition of block being rise in water level and that it leads to force on the bottom which is w + B (reading of electronic scale) . and we know that reading of spring balance is mg - B , the sum of the readings is w+mg (constant) . so the ans must be S. Or is there anything I am getting wrong with the force on bottom thing ?

 

[haruspex]

so the ans must be S

To which, C? The buoyancy is max when h=0 and zero when h>h₀, which fits P.

 

[manasi bandhaokar]

Yes to C. If what I have reasoned is right, the eqn for graph if C should be w+mg . it is independent of B. So it should be a straight line parallel to X axis.

 

[haruspex]

the eqn for graph if C should be w+mg

Do you mean D?

 

[manasi bandhaokar]

Oh no! My bad! I am sorry ! U r right . I meant D. Yeah I am also getting C as P. It's again the messed up origin of paper setters. I think that settles the question. Thanks for all your help everyone. And that for bearing with me.

 

[Abhishek kumar]

Oh no! My bad! I am sorry ! U r right . I meant D. Yeah I am also getting C as P. It's again the messed up origin of paper setters. I think that settles the question. Thanks for all your help everyone. And that for bearing with me.

You must discuss this problem with your teacher and let me know if you find any conclusion.

 

[manasi bandhaokar]

Err... I am not going to any coaching class or college. I have taken a drop for engineering entrance and self studying.

 

[Abhishek kumar]

Err... I am not going to any coaching class or college. I have taken a drop for engineering entrance and self studying.

Its ok. Carry on..

 

[manasi bandhaokar]

Yeah... Thank you.