gabbagabbahey said:Because the arc length is measured from the [itex]\theta=0[/itex] plane (yz plane) to the plane rotated through an angle [itex]d\theta[/itex], So the radius of the arc is the projection of R onto the [itex]\theta=0[/itex] plane which is [itex]Rcos(\theta)[/itex] and so the arc length is [itex]Rcos(\theta)d\theta[/itex].
The center of mass of a hemisphere is the point at which the mass of the hemisphere is evenly distributed in all directions. It is also known as the centroid or the balance point of the hemisphere.
The center of mass of a hemisphere can be calculated by finding the average of all the points within the hemisphere. This can be done by dividing the hemisphere into infinitesimally small pieces and finding the average of each piece's position and mass, and then adding them all together.
Finding the center of mass of a hemisphere is important in various applications, including engineering, physics, and astronomy. It helps determine the stability and balance of objects, predict their motion, and understand their behavior in different environments.
The shape of a hemisphere can affect its center of mass as it determines the distribution of mass within the object. A more symmetrical hemisphere will have its center of mass closer to its geometric center, while an irregularly shaped hemisphere will have its center of mass shifted towards the heavier side.
No, the center of mass of a hemisphere will always be located within the object. This is because the center of mass represents the average position of all the mass within the object, and it cannot be located outside of the object's boundaries.