ilikephysics2
for a spherical object what is its mass in terms of the density of the material, its diameter and any necessary constants?
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lgbasallote
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well volume of a sphere is \[\frac 43 \pi r^3\]
lgbasallote
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and then density = mass/volume
UnkleRhaukus
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\[\rho=\frac mV\]
\[m=V\rho\]
lgbasallote
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so you have \[m = \frac 43\rho \pi r^3\]
lgbasallote
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where \(\rho\) = density
ilikephysics2
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what about its diameter
lgbasallote
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change r to d/2
UnkleRhaukus
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and \(r\) is the radius
(half the diameter)
\(r=\frac d2\)
lgbasallote
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radius = diameter/2 so just change r to d/2
ilikephysics2
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ok so thats the final solution?
lgbasallote
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replace r with d/2....
ilikephysics2
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ok cool and can i give you guys both metals or does just one of you get it?
ilikephysics2
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hey so its (4/3)p(pi)(d/2)^3
lgbasallote
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right
ilikephysics2
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is that right?
UnkleRhaukus
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you can simplify the numbers a bit
ilikephysics2
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unkle...it won't let me give you a metal now
ilikephysics2
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why cant i give it to both of you
UnkleRhaukus
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i dont mind about that
UnkleRhaukus
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\[m=\frac43\rho\pi\left(\frac d2\right)^3\]
use
\[\left(\frac{a}{b}\right)^n=\frac {a^n} {b^n}\]
ilikephysics2
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so it simplifies to 4p(pi)(d/2)?
ilikephysics2
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i crossed out the 3's
lgbasallote
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what he meant was distribute the exponent
UnkleRhaukus
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you can not cancel an index with denominator like that
ilikephysics2
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oh so whats it turn to?
UnkleRhaukus
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\[\left(\frac{a}{b}\right)^n=\frac {a^n} {b^n}\]
\[\left(\frac{d}{2}\right)^3=\frac {d^3} {2^3}\]
ilikephysics2
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thanks :)
UnkleRhaukus
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can you simplify the numbers in the equation now?
ilikephysics2
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yes (4/3)p(pi)(d^3/8) ?
lgbasallote
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you can simplify it further
lgbasallote
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you can multiply the denominators
ilikephysics2
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so 4p(pi)(d/8) ? ?
lgbasallote
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what happened?
ilikephysics2
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can you simplify it?
lgbasallote
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you canceled the denominaotr and teh exponent???
ilikephysics2
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it can't be simplified?
lgbasallote
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it can...you just simplified it wrong....multiply the denominators
lgbasallote
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don't touch anything else
ilikephysics2
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so it (4/24)p(pi)(d^3) ?
UnkleRhaukus
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thats better!
lgbasallote
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yes. now express 4/24 in lowest terms
ilikephysics2
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ok so 1/6 p(pi)(d^3) ? thats final!!!!
lgbasallote
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right!!
lgbasallote
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your turn @UnkleRhaukus
UnkleRhaukus
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\[\color{red}\checkmark\]
UnkleRhaukus
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so the constants necessary to express the volume in terms of diameter are ;
the natural number; 6
the density of the material rho; \(\rho\)
the geometrical constant pi; \(\pi\)
the diameter of the sphere \(d\)
and the dimensional constant \(3\)