Convert from rectangular to spherical coordinates. Cylindrical Coordinates When we expanded the traditional Cartesian coordinate system from two dimensions to three, we simply added a new axis to model the third dimension. In three dimensions, this same equation describes a half-plane. Spherical Coordinates In the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance.
Hint Converting the coordinates first may help to find the location of the point in space more easily. Answer b This set of points forms a half plane. Find the center of gravity of a bowling ball. Determine the velocity of a submarine subjected to an ocean current. Calculate the pressure in a conical water tank. Find the volume of oil flowing through a pipeline. Determine the amount of leather required to make a football.
The origin should be located at the physical center of the ball. Bowling balls normally have a weight block in the center. A submarine generally moves in a straight line. There is no rotational or spherical symmetry that applies in this situation, so rectangular coordinates are a good choice.
The origin should be some convenient physical location, such as the starting position of the submarine or the location of a particular port. A cone has several kinds of symmetry. However, the equation for the surface is more complicated in rectangular coordinates than in the other two systems, so we might want to avoid that choice.
In addition, we are talking about a water tank, and the depth of the water might come into play at some point in our calculations, so it might be nice to have a component that represents height and depth directly. Based on this reasoning, cylindrical coordinates might be the best choice. The orientation of the other two axes is arbitrary. The origin should be the bottom point of the cone. A pipeline is a cylinder, so cylindrical coordinates would be best the best choice. The origin should be chosen based on the problem statement.
It may make sense to choose an unusual orientation for the axes if it makes sense for the problem. A football has rotational symmetry about a central axis, so cylindrical coordinates would work best. The origin could be the center of the ball or perhaps one of the ends. How should we orient the coordinate axes?
Hint What kinds of symmetry are present in this situation? The Cartesian coordinate system allows both positive and negative directions relative to the origin to be specified in each axis. With Cartesian coordinates, each coordinate set defines a unique point in space. The Cartesian coordinate system is often used for straight-line movements, where specifying the motion of an axis is simple — input the location to which the axis should travel or the amount of distance it should travel from the starting point , and it will take a linear path to the specified location.
Although Cartesian coordinates are straightforward for many applications, for some types of motion it might be necessary or more efficient to work in one of the non-linear coordinate systems, such as polar or cylindrical coordinates. For example, if the motion involves circular interpolation, polar coordinates might be more convenient to work in than Cartesian coordinates. Polar coordinates define a position in 2-D space using a combination of linear and angular units.
With polar coordinates, a point is specified by a straight-line distance from a reference point typically the origin or the center of rotation , and an angle from a reference direction often counterclockwise from the positive X-axis.
Recall from above that with Cartesian coordinates, any point in space can be defined by only one set of coordinates. So, if we have a point in cylindrical coordinates the Cartesian coordinates can be found by using the following conversions.
Likewise, if we have a point in Cartesian coordinates the cylindrical coordinates can be found by using the following conversions. In two dimensions we know that this is a circle of radius 5. From the section on quadric surfaces we know that this is the equation of a cone.
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