Probability of object falling into a container

In summary, there is a manufacturing line that moves at 60rpm and has 91 filling locations in one machine cycle. A bolt and nut were found in the bottom of a container with a 29mm diameter opening and a distance of 30cm from the bottom of the machinery. The bolt is 21mm long and threaded through a nut 8mm in diameter. It is uncertain whether the bolt fell from the machine or was manually inserted into the container. There is a probability that the bolt would fall into the container, which could be calculated by comparing the bottle-opening area to the total area. However, there are too many unknown factors to give a precise answer.
  • #1
stlag37
2
0
On a manufacturing line moving uni-directionally at 60rpm (about 825 continers filled per minute) a bolt/nut has been observed in the bottom of a container.

The container neck (opening) is a 29mm diameter. The distance from the bottom of the machinery to the top of the bottle is 30cm. The bolt is 21mm long and is threaded through a nut 8mm in diameter. There are a total of 91 filling locations in one machine cycle.

What is the probability the bolt fell from the machine into the container? Can we hazard a guess as to if it was manually inserted into the container, versus accidentally falling into it?

Thanks!
 
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  • #2
Are you allowed to assume that if the lower tip of the bolt breaks the plane of the imaginary disk at the opening of the bottle, the bolt will definitely drop into, and remain inside of, the bottle?
 
  • #3
Yes, if it fits in the bottle opening (breaking the diametrical plane) it will fall into the bottle and will not be able to, "bounce out."
 
  • #4
About all I can think is maybe you can treat it kind of like a cross-section problem, the way particle physicists do. Can you somehow calculate a ratio of bottle-opening area to total area, and if it is, say, one percent, then you can say there was just a 1% chance that the bolt would land in a bottle by chance? Maybe I am not even on the right track.
 
  • #5
You need some sort of probability distribution for where the bolt would strike. Janitor was suggesting the simplest: the probability that the bolt will fall into the container is the area of the neck of the container divided by the area of the region on which such a bolt could fall. A more accurate distribution might be a 2-dimensional normal distribution about the container. Of course, you would have to include a probability that such a bolt would fall at all. Looks to me like there are simply too many unknowns to give a reasonable result.
 

What is the probability of an object falling into a container?

The probability of an object falling into a container depends on several factors such as the size and shape of the container, the size and shape of the object, and the height at which the object is released. However, assuming all other factors remain constant, the probability can be calculated by dividing the area of the container that the object can fall into by the total area of the container.

How does the size and shape of the container affect the probability?

The size and shape of the container can greatly affect the probability of an object falling into it. A container with a larger opening or a wider mouth has a higher probability of catching the falling object compared to a container with a smaller opening or a narrow mouth. Similarly, a container with a shallower depth or a wider base also has a higher probability of catching the falling object.

What is the role of the height at which the object is released?

The height at which the object is released also plays a significant role in the probability of it falling into a container. The higher the release height, the more time the object has to be affected by external factors such as wind or air resistance, increasing the chances of missing the container. Therefore, the lower the release height, the higher the probability of the object falling into the container.

Can the shape and weight of the object affect the probability?

Yes, the shape and weight of the object can affect the probability of it falling into a container. A heavier object has a higher tendency to fall straight down, while a lighter object is more likely to be affected by external factors and deviate from its trajectory. Similarly, the shape of the object can also affect its trajectory and increase or decrease the chances of it falling into the container.

Is there a way to increase the probability of an object falling into a container?

There are a few ways to increase the probability of an object falling into a container. One way is to have a container with a wider opening and a shallower depth. Another way is to release the object from a lower height. Additionally, using a heavier and more aerodynamic object can also increase the chances of it falling into the container.

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