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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?
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Note: There is no ambient gravitational field.
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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?
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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?
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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?
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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 $A(z).$
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$\mathscr{E} = \displaystyle \int_{0}^{\pi/2} \sqrt{1- \sin(2016 x) }\, dx$
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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