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Q. 46

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
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Short Answer

A small speck of dust with mass has fallen into the hole shown in FIGURE P39.46 and appears to be at rest. According to the uncertainty principle, could this particle have enough energy to get out of the hole? If not, what is the deepest hole of this width from which it would have a good chance to escape?

FIGURE P39.46

The particle does not have enough kinetic energy that could transform into potential energy and escape the hole. The deepest hole from which the speck could have a good chance of escaping is .

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Step by Step Solution

Step 1 Given Information A small speck of dust with mass   has fallen into the hole and appears to be at rest 

The uncertainty in knowing the position of the dust speck inside a hole of a width of is . Now, the uncertainty in the velocity of the particle can be calculated using the Heisenberg uncertainty principle

Assuming the most accurate measurements possible, we can replace with . Moreover, we know that . Thus

which means that the range of possible velocities for the dust speck is from to . In terms of speed, the maximum possible speed for the dust speck is . Thus, the maximum kinetic energy of the speck is

Now, in order for the speck to get out of the hole, it must gain potential energy that allows it to reach a height of , and this amount of potential energy can be calculated as follows

Given Information :The potential energy the dust speck needs to get out of the hole is much larger than the available kinetic energy, so the particle does not have enough kinetic energy that could transform into potential energy and escape the hole. Now, to find the deepest hole from which the speck could have a good chance of escaping we need to equate the kinetic energy with the potential energy and find the value of h                                                               

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