How far will sphere fall into a paraboloid? (uses curvature I think )

In summary, the problem involves dropping a sphere with a radius of 4 into a paraboloid and determining the distance between the bottom point of the sphere and the bottom point of the paraboloid when the sphere stops falling. The second part of the problem asks for the radius of the largest sphere that can fall all the way and touch the bottom of the paraboloid. To solve this problem, one needs to compare the curvatures of the paraboloid and the sphere and find the point where they are equal, which would be the "pinch point" where the sphere stops falling. The solutions for the two parts of the problem are 12.25 and 0.50 respectively, and can be obtained by comparing the
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Homework Statement


A sphere with a radius of 4 is dropped into a paraboloid. How far is the bottom point of the sphere from the bottom point of the paraboloid when the sphere stops falling? What is the radius of the largest sphere that will fall all of the way and touch the bottom of the paraboloid (distance=0)?

Homework Equations


Paraboloid (fixed location with nubby bit touching origin)
z=x2+y2

Sphere (falls with gravity until it gets stuck, center unknown)
x2+y2+z2=42=16

The Attempt at a Solution


The was an optional "challenge" problem and the solutions from the back of the book are (to two decimal places) 12.25 and 0.50 respectively. This comes from the chapter about curvature so I'm pretty sure it involves comparing the curvature of the "container" paraboloid with the sphere that you "drop in". Since the sphere has no fixed center it seems to me that you need to compare the curvatures. My intuition is that there is a circle around the paraboloid that will be a "pinch point" and stop the sphere from falling anymore. At such a point the paraboloid and sphere would have the same curvature...I think. I can't figure out how to take it any farther than that.
 
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I'm really just looking for an explanation of the solution so that I can understand it. Thanks very much!
 

Related to How far will sphere fall into a paraboloid? (uses curvature I think )

1. How does the curvature of a paraboloid affect the distance a sphere will fall?

The curvature of a paraboloid plays a significant role in determining the distance a sphere will fall. The greater the curvature, the steeper the slope of the paraboloid and the shorter the distance the sphere will fall. On the other hand, a flatter paraboloid with less curvature will result in a longer distance for the sphere to fall.

2. Is there a mathematical equation to calculate the distance a sphere will fall into a paraboloid?

Yes, there is a mathematical equation that can be used to calculate the distance a sphere will fall into a paraboloid. It involves using the radius of the sphere, the height of the paraboloid, and the curvature of the paraboloid. This equation is known as the geodesic equation and is commonly used in differential geometry.

3. What other factors besides curvature can affect the distance a sphere will fall?

Besides curvature, the weight and size of the sphere can also affect the distance it will fall into a paraboloid. A heavier or larger sphere will have more momentum and will likely travel a shorter distance compared to a lighter or smaller sphere. Additionally, air resistance and the surface of the paraboloid can also impact the distance the sphere will fall.

4. Can a sphere fall infinitely into a paraboloid?

No, a sphere cannot fall infinitely into a paraboloid. The distance the sphere will fall is limited by the height of the paraboloid. Once the sphere reaches the bottom of the paraboloid, it will come to a stop. However, if the paraboloid has a circular opening at the bottom, the sphere may continue to roll out of the paraboloid and fall further.

5. How is the distance a sphere falls into a paraboloid relevant in real-world applications?

The concept of a sphere falling into a paraboloid can be applied in various real-world scenarios, such as in physics experiments, engineering designs, and even sports. For example, in roller coasters, the shape of the track is often modeled as a paraboloid to control the speed and trajectory of the carts. Understanding the distance a sphere will fall into a paraboloid can also help in predicting the path of a projectile and designing optimal landing zones for safe landings.

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