# Problems of the Week

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The center of a solid sphere of radius $$R$$ is located a distance of $$2R$$ from a point-particle.

Approximately what percentage of the gravitational force felt by the point-particle is due to the blue half of the sphere?


Note: There is no ambient gravitational field.

Danielle, Lilliana, and Melody play a fighting video game in tournament mode. In this mode, two players play a match, and the winner of the match plays a new match against the player who was sitting out. This continues until a player wins two matches in a row. Danielle and Lilliana play the first match.

If they are all equally skilled at the game, then what is the probability that Melody will win the tournament?

A variable capacitor consists of two metal semicircles of radius $$R$$ with vertical separation $$d.$$ The capacitor is charged when $$\varphi=0,$$ and is then disconnected from the voltage source. The discs are then rotated through the angle $$\varphi = 90^\circ.$$

How does the rotation change the energy that's stored in the capacitor?

The height profile of a valley basin can be described by the two-dimensional parabolic function$h(x, y) = \frac{x^2}{144\,\text{m}} + \frac{y^2}{324\,\text{m}}.$ Now the basin is filled by a rainstorm to a height of $$h_0 = 12\,\text{m}.$$ What is the volume of the resulting lake $$($$in $$\text{m}^3)$$ to the nearest integer?



Hint: Find the shape of the cross-sectional areas enclosed by the equipotential lines $$h(x, y) = z = \text{constant}$$. The volume then results from the integral $$\displaystyle V = \int_0^{h_0} A (z)\, dz$$ over the cross-sectional area $$A(z).$$

$\mathscr{E} = \displaystyle \int_{0}^{\pi/2} \sqrt{1- \sin(2016 x) }\, dx$

Find $$\mathscr{E}.$$

Inspiration

Bonus: Generalize for $$\displaystyle \int_{0}^{\pi/2} \sqrt{1- \sin(12n x) }\, dx$$, where $$n$$ is a positive integer, and prove it.

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