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Spherical coordinates watch

1. Calculate the z-component of the centre of mass for a northern hemisphere of radius with constant density using spherical coordinates defined by:

For a solid with density occupying a region ,

where

I have the solution but I'm wondering why the limits of is for calculating M then it changes to when calculating ?

.

.
2. I'm guessing that it's a mistake; it doesn't look like they've done anything other than multiply the integrand by , so it shouldn't affect the limits in any way.
3. (Original post by nuodai)
I'm guessing that it's a mistake; it doesn't look like they've done anything other than multiply the integrand by , so it shouldn't affect the limits in any way.
So I'm guessing both integrals should have limits for a 'northern hemisphere'?
4. (Original post by TheEd)
So I'm guessing both integrals should have limits for a 'northern hemisphere'?
No, they should both be . If P is a point in then tells you the angle that OP makes with the positive z-axis, so if it went up to then you'd have a full sphere.
5. (Original post by nuodai)
No, they should both be . If P is a point in then tells you the angle that OP makes with the positive z-axis, so if it went up to then you'd have a full sphere.
Yeah makes sense now. Ta.

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