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ilikephysics2

  • 2 years ago

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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  1. lgbasallote
    • 2 years ago
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    well volume of a sphere is \[\frac 43 \pi r^3\]

  2. lgbasallote
    • 2 years ago
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    and then density = mass/volume

  3. UnkleRhaukus
    • 2 years ago
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    \[\rho=\frac mV\] \[m=V\rho\]

  4. lgbasallote
    • 2 years ago
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    so you have \[m = \frac 43\rho \pi r^3\]

  5. lgbasallote
    • 2 years ago
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    where \(\rho\) = density

  6. ilikephysics2
    • 2 years ago
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    what about its diameter

  7. lgbasallote
    • 2 years ago
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    change r to d/2

  8. UnkleRhaukus
    • 2 years ago
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    and \(r\) is the radius (half the diameter) \(r=\frac d2\)

  9. lgbasallote
    • 2 years ago
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    radius = diameter/2 so just change r to d/2

  10. ilikephysics2
    • 2 years ago
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    ok so thats the final solution?

  11. lgbasallote
    • 2 years ago
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    replace r with d/2....

  12. ilikephysics2
    • 2 years ago
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    ok cool and can i give you guys both metals or does just one of you get it?

  13. ilikephysics2
    • 2 years ago
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    hey so its (4/3)p(pi)(d/2)^3

  14. lgbasallote
    • 2 years ago
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    right

  15. ilikephysics2
    • 2 years ago
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    is that right?

  16. UnkleRhaukus
    • 2 years ago
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    you can simplify the numbers a bit

  17. ilikephysics2
    • 2 years ago
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    unkle...it won't let me give you a metal now

  18. ilikephysics2
    • 2 years ago
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    why cant i give it to both of you

  19. UnkleRhaukus
    • 2 years ago
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    i dont mind about that

  20. UnkleRhaukus
    • 2 years ago
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    \[m=\frac43\rho\pi\left(\frac d2\right)^3\] use \[\left(\frac{a}{b}\right)^n=\frac {a^n} {b^n}\]

  21. ilikephysics2
    • 2 years ago
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    so it simplifies to 4p(pi)(d/2)?

  22. ilikephysics2
    • 2 years ago
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    i crossed out the 3's

  23. lgbasallote
    • 2 years ago
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    what he meant was distribute the exponent

  24. UnkleRhaukus
    • 2 years ago
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    you can not cancel an index with denominator like that

  25. ilikephysics2
    • 2 years ago
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    oh so whats it turn to?

  26. UnkleRhaukus
    • 2 years ago
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    \[\left(\frac{a}{b}\right)^n=\frac {a^n} {b^n}\] \[\left(\frac{d}{2}\right)^3=\frac {d^3} {2^3}\]

  27. ilikephysics2
    • 2 years ago
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    thanks :)

  28. UnkleRhaukus
    • 2 years ago
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    can you simplify the numbers in the equation now?

  29. ilikephysics2
    • 2 years ago
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    yes (4/3)p(pi)(d^3/8) ?

  30. lgbasallote
    • 2 years ago
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    you can simplify it further

  31. lgbasallote
    • 2 years ago
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    you can multiply the denominators

  32. ilikephysics2
    • 2 years ago
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    so 4p(pi)(d/8) ? ?

  33. lgbasallote
    • 2 years ago
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    what happened?

  34. ilikephysics2
    • 2 years ago
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    can you simplify it?

  35. lgbasallote
    • 2 years ago
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    you canceled the denominaotr and teh exponent???

  36. ilikephysics2
    • 2 years ago
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    it can't be simplified?

  37. lgbasallote
    • 2 years ago
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    it can...you just simplified it wrong....multiply the denominators

  38. lgbasallote
    • 2 years ago
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    don't touch anything else

  39. ilikephysics2
    • 2 years ago
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    so it (4/24)p(pi)(d^3) ?

  40. UnkleRhaukus
    • 2 years ago
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    thats better!

  41. lgbasallote
    • 2 years ago
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    yes. now express 4/24 in lowest terms

  42. ilikephysics2
    • 2 years ago
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    ok so 1/6 p(pi)(d^3) ? thats final!!!!

  43. lgbasallote
    • 2 years ago
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    right!!

  44. lgbasallote
    • 2 years ago
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    your turn @UnkleRhaukus

  45. UnkleRhaukus
    • 2 years ago
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    \[\color{red}\checkmark\]

  46. UnkleRhaukus
    • 2 years ago
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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 \(3\)

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