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2017-11-27 Advanced


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 \(\mathscr{E}.\)


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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