Chapter 3: Problem 22
Use spherical polar coordinates \(r, \theta, \phi\) to find the CM of a uniform solid hemisphere of radius R, whose flat face lies in the \(x y\) plane with its center at the origin. Before you do this, you will need to convince yourself that the element of volume in spherical polars is \(d V=r^{2} d r \sin \theta d \theta d \phi\). (Spherical polar coordinates are defined in Section 4.8. If you are not already familiar with these coordinates, you should probably not try this problem yet.)
Short Answer
Step by step solution
Understand the Volume Element
Set Up the Integral for Mass
Calculate the Mass
Find the Center of Mass in Spherical Coordinates
Calculate the Center of Mass Coordinate
Summarize the Coordinates of the Center of Mass
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Spherical Coordinates
- \(r\) denotes the radial distance from the origin, extending outwards. - \(\theta\) is the polar angle measured down from the positive \(z\)-axis. - \(\phi\) is the azimuthal angle around the \(z\)-axis, similar to longitude.
This system of coordinates is particularly useful for problems involving spheres or spherical symmetry, such as finding the center of mass of a hemisphere. By employing these coordinates, one can easily account for changes in dimensions that occur as you move further from the origin, particularly when integrating over volumes.
Hemisphere
- The hemisphere extends from the bottom plane upward to its curved surface. - Symmetry along the \(z\)-axis helps in reducing the complexity of mathematical computation, especially while determining the center of mass.
The center of mass for this shape must factor in its geometric symmetry, particularly because any shifts in density would affect its location. For a uniform hemisphere, the symmetrical nature around the vertical axis means that the center of mass lies on this axis.
Volume Element
This expression derives from how distances and areas stretch when moving away from the central point.
- The factor \(r^2\) represents the radial scaling as you move outward from the origin.
- \(\sin \theta\) accounts for the variation in distance along the axis at different \(\theta\) angles.
It's important to remember this when setting up integrals, as incorrect volume element expressions can lead to erroneous results. To calculate physical properties like mass or center of mass, integrating over the entire volume element correctly is crucial.
Integration in Polar Coordinates
Here’s how you would typically set it up for a uniform solid hemisphere:
- Start by integrating over \(\phi\), as it typically spans from 0 to \(2\pi\) and accounts for the full circle around the \(z\)-axis.
- Next, the integral over \(\theta\) goes from 0 to \(\pi/2\), reflecting the dome-like shape of the hemisphere.
- Finally, the integral over \(r\) spans from 0 to the hemisphere's radius \(R\).
The order and limits of integration are determined by the physical shape and how it is oriented in space. When properly handled, this integration helps find quantities such as mass and the precise position of the center of mass.