Center of mass of solid hemisphere by using angle as variable.

In summary, the conversation involved a discussion about solving for the center of mass of a solid hemisphere using the method of using the angle as a variable. However, the answer did not match and the person was seeking help in finding their mistake. They also provided their attempt at a solution and later realized their mistake.
  • #1
AlchemistK
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0

Homework Statement


I've just started with center of mass, and instead of the method in the book, I tried solving the center of mass for a solid hemisphere using angle as a variable, but the answer didn't match.
A small disk of mass "dm" is taken, which subtends an angle "dθ" at the center.The total mass of the hemisphere is "M". Please refer to the attachment for the figure and my full attempt.
Where have i gone wrong?

Homework Equations



y(cm) = ∫y dm / ∫ dm = 1/M (∫y dm)

The Attempt at a Solution


Please refer to the attachment.
 

Attachments

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  • #2
Mod Note: moved to calculus & beyond. Please post in the correct forum next time!
 
  • #3
Ah. Never mind, I found my mistake. Very idiotic of me.
 

What is the definition of center of mass?

The center of mass of an object is the point where the mass of the object is evenly distributed in all directions.

How is the center of mass of a solid hemisphere calculated using angle as a variable?

The center of mass of a solid hemisphere can be calculated using the following formula: x = 0, y = 0, z = (3R/8) * sin(theta), where R is the radius of the hemisphere and theta is the angle measured from the base of the hemisphere.

What is the significance of using angle as a variable in calculating the center of mass of a solid hemisphere?

Using angle as a variable allows for a more precise calculation of the center of mass, as it takes into account the varying distribution of mass at different angles from the base of the hemisphere.

Can the center of mass of a solid hemisphere be located outside of the object?

No, the center of mass of a solid hemisphere will always be located within the object, as it is the point where the mass is evenly distributed in all directions.

How is the center of mass of a solid hemisphere affected by changes in the radius of the hemisphere?

The center of mass of a solid hemisphere will shift as the radius of the hemisphere changes. As the radius increases, the center of mass will move further away from the base of the hemisphere along the z-axis.

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