## ilikephysics2 3 years ago for a spherical object what is its mass in terms of the density of the material, its diameter and any necessary constants?

1. lgbasallote

well volume of a sphere is $\frac 43 \pi r^3$

2. lgbasallote

and then density = mass/volume

3. UnkleRhaukus

$\rho=\frac mV$ $m=V\rho$

4. lgbasallote

so you have $m = \frac 43\rho \pi r^3$

5. lgbasallote

where $$\rho$$ = density

6. ilikephysics2

7. lgbasallote

change r to d/2

8. UnkleRhaukus

and $$r$$ is the radius (half the diameter) $$r=\frac d2$$

9. lgbasallote

radius = diameter/2 so just change r to d/2

10. ilikephysics2

ok so thats the final solution?

11. lgbasallote

replace r with d/2....

12. ilikephysics2

ok cool and can i give you guys both metals or does just one of you get it?

13. ilikephysics2

hey so its (4/3)p(pi)(d/2)^3

14. lgbasallote

right

15. ilikephysics2

is that right?

16. UnkleRhaukus

you can simplify the numbers a bit

17. ilikephysics2

unkle...it won't let me give you a metal now

18. ilikephysics2

why cant i give it to both of you

19. UnkleRhaukus

20. UnkleRhaukus

$m=\frac43\rho\pi\left(\frac d2\right)^3$ use $\left(\frac{a}{b}\right)^n=\frac {a^n} {b^n}$

21. ilikephysics2

so it simplifies to 4p(pi)(d/2)?

22. ilikephysics2

i crossed out the 3's

23. lgbasallote

what he meant was distribute the exponent

24. UnkleRhaukus

you can not cancel an index with denominator like that

25. ilikephysics2

oh so whats it turn to?

26. UnkleRhaukus

$\left(\frac{a}{b}\right)^n=\frac {a^n} {b^n}$ $\left(\frac{d}{2}\right)^3=\frac {d^3} {2^3}$

27. ilikephysics2

thanks :)

28. UnkleRhaukus

can you simplify the numbers in the equation now?

29. ilikephysics2

yes (4/3)p(pi)(d^3/8) ?

30. lgbasallote

you can simplify it further

31. lgbasallote

you can multiply the denominators

32. ilikephysics2

so 4p(pi)(d/8) ? ?

33. lgbasallote

what happened?

34. ilikephysics2

can you simplify it?

35. lgbasallote

you canceled the denominaotr and teh exponent???

36. ilikephysics2

it can't be simplified?

37. lgbasallote

it can...you just simplified it wrong....multiply the denominators

38. lgbasallote

don't touch anything else

39. ilikephysics2

so it (4/24)p(pi)(d^3) ?

40. UnkleRhaukus

thats better!

41. lgbasallote

yes. now express 4/24 in lowest terms

42. ilikephysics2

ok so 1/6 p(pi)(d^3) ? thats final!!!!

43. lgbasallote

right!!

44. lgbasallote

45. UnkleRhaukus

$\color{red}\checkmark$

46. UnkleRhaukus

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 $$3$$