document.write( "Question 1198114: A beverage container is made of aluminum (SG = 2.7) with a thickness of 0.1 mm.
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\n" ); document.write( "The diameter is 64 mm, the height is 128 mm.
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\n" ); document.write( "How much water (SG = 1) will produce the lowest center of gravity? \n" ); document.write( "

Algebra.Com's Answer #831638 by ikleyn(52802) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "A beverage container is made of aluminum (SG = 2.7) with a thickness of 0.1 mm.
\n" ); document.write( "It's cylindrical, with a top and bottom of Al.
\n" ); document.write( "The diameter is 64 mm, the height is 128 mm.
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\n" ); document.write( "How much water (SG = 1) will produce the lowest center of gravity?
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\n" ); document.write( "\n" ); document.write( "First, SG means \"specific gravity\".\r
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\n" ); document.write( "\n" ); document.write( "Second, this problem is, OBVIOUSLY, an entertainment problem to entertain your brain/mind.\r
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\n" ); document.write( "\n" ); document.write( "Therefore, I will give you the basic ideas, only, leaving all computations to you.\r
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\n" ); document.write( "\n" ); document.write( "Third, I will tell you, which parameters you may neglect when you make your calculations.\r
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document.write( "    You should use the thickness of the container's wall 0.1 mm, when you calculate the mass \r\n" );
document.write( "    of the container, but you can neglect this thickness when you calculate the volume of the water \r\n" );
document.write( "    inside the container.\r\n" );
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\n" ); document.write( "\n" ); document.write( " Now about the solution itself.\r
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document.write( "(a)  First consider the empty container (with no water inside).\r\n" );
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document.write( "     Then it is clear that the center of gravity is the center of this empty cylinder.\r\n" );
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document.write( "     So, the position of the center of gravity of the empty container is half \r\n" );
document.write( "     the height of the container.\r\n" );
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document.write( "     The mass of the container you can easily calculate as the mass of the empty cylinder\r\n" );
document.write( "     with the given dimensions and the SG of aluminum.\r\n" );
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document.write( "     So, I suppose that you calculate this mass  of the container on your own.\r\n" );
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document.write( "     You just know that the mass   of the empty container is located at the height \r\n" );
document.write( "     H/2, where H is the height of the container.\r\n" );
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document.write( "(b)  Second, when you fill the water to the height h of the container, the center of the gravity \r\n" );
document.write( "     of water is at the height h/2 and the mass of water  you can compute easily\r\n" );
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document.write( "                   = .\r\n" );
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document.write( "(c)  So, you have the mass  concentrated at the height H/2,\r\n" );
document.write( "         and you have the mass of water  concentrated at the height h/2.\r\n" );
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document.write( "     OBVIOUSLY, the mass of the combined system is   and the height \r\n" );
document.write( "     of center of mass of the combined system is  .    (1)\r\n" );
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document.write( "     Notice that this term,  is zero when h= 0 (then  is zero)\r\n" );
document.write( "     and  is zero when h = H, so at these two values of h\r\n" );
document.write( "     the gravity center is at half of the height H at h= 0 or h= H.\r\n" );
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document.write( "(d)  After that, your task is to combine everything together in formula (1).\r\n" );
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document.write( "     You will have the position of the center as a quadratic function of h.\r\n" );
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document.write( "     To get the minimum of the center of mass height, you find the minimum of this function on h.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Having these instructions, boldly go forward and have fun (!)\r
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\n" ); document.write( "\n" ); document.write( "I hope that your level is adequate to the problem level (otherwise, what is the reason for you to try ?)\r
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