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# Problem of the Month

Do you want to win a prize and score bragging rights for a month? Are you a first or second year student preparing for quals and want some more practice? Are you a senior student looking to brush up on some physics outside of what you usually do? Or maybe you’re the kind of person who sees a good problem and can’t possibly continue on with your day without solving it? Whatever your reason, you’ve come to the right place. Check out our most recent problem of the month below!

Prizes: Past prizes include \$20 gift card to Zingerman's Deli, \$20 gift card to Literarti Books and your choice in pizza toppings at the AP Seminar.

Submission Info: Submit a PDF of your solution to: APProblemOfTheMonth@umich.edu. The submission deadline is always the last day of the month, by 11:59pm. Typed solutions are preferred, but handwritten solutions will also be accepted if they're legible. The winning solution will be chosen by Prof. Kurdak shortly thereafter. Solutions will be judged on their correctness, clarity, thoroughness, and creativity.

Other Information: Have any questions about the problem? Ask them at APProblemOfTheMonth (at) umich.edu. This contest is open only to AP students. You can work with other students if you'd like, but please indicate that in your submission. Do you have other problems you'd like to suggest that we use in the future? Email us them too!

### April 2017

Brick Stacking:

How far can you stack bricks over an edge? See the here or here for a more visual explanation (or this video for the inspiration for this problem).

All the bricks are the same size and weight, and are all perfectly rectangular, with length L. There is no adhesive - all the bricks must simply be balanced with gravity. And you can only place 1 brick per row. Given N bricks, how far can you make the top-most brick go over the edge? And, more interestingly, as N goes to infinity, to what (if anything) does that distance-over-the-edge converge?

Extra credit - Anything you can think to add! Maybe build one yourself!**

**Besides being a fun math problem, it's also fun to build this. Even when using something heavy, like a stack of bricks, when the tower is nearly perfectly balanced, it's counter-intuitively 'light' to the touch (see 2:55 in that video). Understanding the physics of that isn't hard, but being able to keep a giant stack of bricks from falling over by gentling pushing them up with your pinky is still quite strange feeling.

Winner/Runner Up:

Prize: \$20 Gift Card to Literati Book StoreWe are still accepting submissions!

### March 2017

Earth’s Magnetic Field

The Earth's magnetic field will flip one day (perhaps soon?) but it won't do so instantaneously - and so there will be some intermediate periods where the magnetic field is effectively gone, resulting in increased radiation (charged particles) from the Sun hitting the Earth. The lack of a magnetic field won't threaten the survival of humans, but they are likely to increase incidences of cancer, may slowly strip away the Earth's atmosphere, and may damage satellites and power supplies (at least according to Wikipedia).

So suppose it's the not-too-distant future where the Earth's magnetic field is disappearing. To save the planet, you want to see if it's feasible to create an artificial magnetic field that mimics the Earth's current magnetic field. In this problem, you'll work through a sort of first approximation to see what it might take to make this happen, and see whether or not it even seems physically possible.

Part 1: Given that the average magnetic field on the surface of the Earth is around 45 microTesla (here), find the effective magnetic dipole moment.

Part 2: If a loop of wire encircling the Earth's equator were used to try to reproduce this magnetic dipole moment, how much current would it take?

Part 3: To drive a current like this, you'd need a lot of power to overcome the ohmic resistance. If all of the world's electrical power generation used today (around 20 trillion kWhr/yr, here) were used to drive this current, what would the maximum resistance need to be?

Part 4: What would be the diameter of this wire if it were made out of copper (conductivity = 17 nano-ohm-meters, here)? And how much copper would this take? Compare that to the global reserves of copper (700 million tonnes, here), or the total amount of copper ever mined (935 million tons, estimated from here), or the total amount of copper even on the planet (2 quadrillion tonnes, estimated from here, here, and here). So, would it even be physically possible?

Extra Credit: A ring of current only looks like a magnetic dipole from far away. Can you derive either the exact equations for the magnetic field around a ring of current, or come up with an approximation of them beyond just the dipole approximation?

Winner: Joseph Levesque

Prize: \$20 gift card to Zingerman's Deli

### February 2017

Solar Sail

In order to impress your (future) significant other for Valentine's Day, you decide to capture the most romantic photo you can think of: a picture of Pluto’s heart-shaped crater. While the New Horizons space probe captured a beautiful photo, it did it in a paltry 1 MP resolution*. You want to capture it in much higher resolution for your Valentine, and so you decide to send a GoPro to Pluto so you can get a 12MP photo, or even better, a 4K video, of Pluto’s heart-shaped crater. The problem is, you don’t have the full resources of NASA to accomplish your task – all you get is a CubeSat, a small box that launches to space aboard a scheduled rocket launch. And in this CubeSat, you decide you’ll pack a solar sail, much like this one created by Bill Nye's Planetary Society. The task this month is to figure out whether or not a solar sail will get your camera to Pluto, and if so, determine how long it will take.

A solar sail can be quite interesting – while a rocket can reach very high speeds, once it runs out of fuel, the Sun’s gravitational force is always pulling it inwards, slowing it down. A solar sail, on the other hand, is constantly being hit with photons from the Sun, so, in theory, it would always be receiving a force which (hopefully) could be designed to exceed the gravitational force, and therefore always be accelerating. So you’d expect over short distances that a rocket would be the fastest mode of transportation, but that over long distances, a solar sail might be better.

Assumptions/Simplifications: Your GoPro weighs 118 grams, and your CubeSat is capped at 1.3 kg. Assume that the remaining weight available to you is spent purely in solar sail material – in this ideal world, your solar sail is able to magically assemble itself in space, and requires no other support/machinery/communications equipment. Furthermore, assume the starting position of the sail is at 1 AU from the Sun, and that your initial velocity is zero – in other words, once your CubeSat deploys the sail/GoPro, for that first instant of time, it is motionless with respect to the Sun. Consider only 1D motion traveling radially away from the Sun, and ignore any other sources of gravity, like Earth/Pluto/Jupiter/etc - so no gravity assists or anything complicated like that.

Part 1 – Assume that all of the sun’s photons are monochromatic (say 500nm) and that your solar sail is perfectly 100% reflective. Given the total radiation power of the Sun (382.8 YW), find an expression for the solar radiation pressure as a function of distance from the Sun. Does this expression change if you consider the fact that the Sun is actually a black body, radiating a wide spectrum of frequencies instead of just a single frequency? Remember that photon momentum scales with frequency.

Part 2 – Find an expression for the critical (maximum) areal density (sail mass / sail area) for a completely empty solar sail (i.e. without the GoPro attached). If such a solar sail were made of Mylar (density 1390 kg/m^3), how thick would it need to be? Compare to the thickness of a typical Mylar balloon (about 14 microns).

Part 3 – For our sail, let's use a very idealized sail made of ultra-thin aluminum (20nm thick, density 2700 kg/m^3). Given the mass constraints, what area solar sail does this allow, and compare that area to something tangible to give it context. Is the effective areal density of this combined sail/GoPro below the critical areal density from Part 2?

Part 4 – Now find (and solve) the equations of motion. How long does it take to get to Pluto (39.5 AU)? And how does this compare to the time it took New Horizons to get there (3,463 days)?

Extra Credit – Any other modifications to the problem you can think of!*Technically, while the camera on board captures only 1024x1024 pixels, it took a bunch of pictures and stitched them together for a shot of Pluto at least 8000x8000 pixels (61 MP). More awesome pictures of Pluto here

Winner / Runner-Up: No February winner

### January 2017

Faster Than Free Fall

There are two dumbbells held at the same height from the ground. One is completely free, and the other has a chain attached to it, where the other end of the hanging chain is fixed at the starting height. If both dumbbells are dropped at the same time, they hit the ground at almost the same time, but if the experiment is done carefully enough, the dumbbell with the chain actually hits the ground first. How is it possible that the chained dumbbell falls faster than free fall? [See the last few minutes of the National Geographic television show “Street Genius” Season 2, Episode 12 for the inspiration for this problem].

• What assumptions are you making in creating your model? If you can, explain why those assumptions are justified.
• What if a rope were used instead of a chain?
• What parameters affect the time difference between the two objects falling? Find how that time difference varies, quantitatively, and take some limits to understand the problem (in the limit that the chain disappears, the time difference should clearly be zero).
• Suppose you were designing this experiment yourself, and you wanted to have a more substantial difference in time so you didn't need to rely on a slow motion video replay to convince someone of the difference. Can you find what parameters you would need to choose in order to make the chain arrive at the ground 0.1 seconds faster than free fall? 0.5 sec? 1.0 sec?
• Given what you've learned about your model and its parameters, does the explanation given in the video agree with your analysis? Does it disagree? Would your explanation to someone who's not a physics student have been any different?
• This problem is fairly contrived - after all, we don't drop weights and chains from cherry pickers all that often. But are there other more practical applications where similar physics become important?

Anything else you can think to add or modify to the problem?

Winner: Thomas Shaw

Runners Up: Joseph Levesque

Prize: Gyro meat, tomatoes, and feta cheese pizza

### November 2016

Ant on a Rubber Band

An ant is positioned on a circular rubber band of diameter D. In time step T, the ant walks a distance d (<πD). After each time step, the rubber band is stretched to increase the diameter by an additional D (and the rubber band conveniently never breaks). Does the ant ever make it all the way around? If so, how long does it take? For example, consider the values of D=1m, d=1cm, and T=1 sec.

Extra Credit 1: Find the largest value for D such that the ant is able to complete the trip before dying, given d=1cm, T=1sec, and that the average ant lives for 60 days (and that it begins walking the moment its born).

Extra Credit 2: Repeat the problem with its continuous analogue. In other words, the diameter increases linearly and smoothly in time, and the ant travels smoothly at a constant speed. Does the ant ever make it all the way around, and if so, how long does it take?

Extra Credit 3: Can you think of any other interesting ways to modify the problem?

Winner: Eric Martin

Runners Up: Albert Liu, Everardo Olide

Prize: BBQ Chicken Pizza