January 26th, 2013, 06:43 PM  #1 
Senior Member Joined: Dec 2012 Posts: 450 Thanks: 0  Unequal Weights
I have a weghing balance.But when nothing is placed at both ends it droops down to one end. I weighed something using this.It weighed 100 grams on one side and 144 grams on the other side. Can you figure out the real weight of the object? This is a real world problem that i observed recently.And i solved it. Can anybody do what i did?Give me an explanation too. 
January 26th, 2013, 07:22 PM  #2 
Member Joined: Jul 2012 Posts: 60 Thanks: 0 Math Focus: Calculus  Re: UNEQUAL WEIGHTS
Did you mean the [color=#8000BF]same object [/color]weighs 100 grams on one side and 144 on the other? Since you said it's a real world problem, a bit more information is needed like: what's wrong with the weighing balance? Is one plate heavier than the other, is it not leveled, or is one of the bars longer than the other? I'll just assume you meant that one plate is heavier than the other. Then the mass of the object you are weighing is 100 grams. Why? Let's first see how a weighing balance works: You place an object on one side, and then you have to place a metal weight on the other side until it's balanced. When it's balanced, you read the number the scale is showing. Well let's just say plate 1 is heavier than plate 2, and the difference between them is If you place your object on plate 1, then you will have to place a weight of onto the other side, so you will be measuring the mass of your object + the additional mass of the heavier plate = 144 If you place your object on plate 2, (assuming your object is heavier than ) you will have to place a weight of onto the other side. So you will successfully be weighing your object and getting a reading of 100 grams. 
January 26th, 2013, 09:32 PM  #3 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,260 Thanks: 198  Re: Unequal Weights
I have a different approach. Assuming 1 end is heavier, the fulcrum exactly in the middle, and everything else about the balance is uniform, homogeneous and isotropic... you can say let m = mass of your object, let x = additional weight at one end, then, m + x = 144 m  x = 100 m = 122 
January 30th, 2013, 05:24 PM  #4  
Senior Member Joined: Dec 2012 Posts: 450 Thanks: 0  Re: Unequal Weights Quote:
But your solution is very clear and simple,i appreciate that.  
January 30th, 2013, 05:25 PM  #5  
Senior Member Joined: Dec 2012 Posts: 450 Thanks: 0  Re: UNEQUAL WEIGHTS Quote:
 
January 30th, 2013, 09:08 PM  #6 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,260 Thanks: 198  Re: Unequal Weights
We can use archimede's law of the lever and center of gravity idea, explained wonderfully by julius sumner miller in the links below. http://www.youtube.com/watch?v=vYbBakL9Cj8 http://www.youtube.com/watch?v=3CBuoe9iCQI The weight of the object in question is the same for both weighings. Whithout loss of generality, let the length of the ballance be 1 and the distance of the fulcrum from the short end be D. Then according to the law of the lever. (1) (2) Expand (1) and (2) (3) (4) add (3) and (4) (5) Substitute (5) in (1) (6) rearrange (6) (7) the positive solution of (7) is D = 5/11 , plug this into (2) W = 120 P.S. I hope you have an easier way. [edit] We could have plugged 5/11 into (5) but it's also good to plug into (2) in order to come full circle so to speak. [endedit] 
January 31st, 2013, 05:44 AM  #7 
Member Joined: Jul 2012 Posts: 60 Thanks: 0 Math Focus: Calculus  Re: Unequal Weights
Well did I do it right or what? Is that how you solved it? 
January 31st, 2013, 07:09 AM  #8  
Member Joined: Jul 2012 Posts: 60 Thanks: 0 Math Focus: Calculus  Re: Unequal Weights Quote:
Here you go: M is moment of force btw The circle is your object In order to stay leveled, the moment of force of the left side and the right side need to be equal.  
January 31st, 2013, 05:12 PM  #9  
Senior Member Joined: Dec 2012 Posts: 450 Thanks: 0  Re: UNEQUAL WEIGHTS Quote:
m1m0=100 so you get m1=122 You have done it wrong.....i think....  
January 31st, 2013, 05:51 PM  #10  
Senior Member Joined: Dec 2012 Posts: 450 Thanks: 0  Re: Unequal Weights Quote:
Yes i have an easier way i think....But i doubt if it is the right way after seeing how you guys do it. Suppose the lengths of the arms are a and b units and the weight of the object X grams. a*X=b*100[* is the symbol for multiplication]>eq (1)[i dont have a good explanation for this] a*144=b*X>eq (2) eq(1)/eq(2)gives X/144=100/X So X^2=100*144 which gives X=120 grams. Am i right?  

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