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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.
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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 φ=0, and is then disconnected from the voltage source. The discs are then rotated through the angle φ=90∘.
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 functionh(x,y)=144mx2+324my2. Now the basin is filled by a rainstorm to a height of h0=12m. What is the volume of the resulting lake (in m3) to the nearest integer?
Hint: Find the shape of the cross-sectional areas enclosed by the equipotential lines h(x,y)=z=constant. The volume then results from the integral V=∫0h0A(z)dz over the cross-sectional area A(z).
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E=∫0π/21−sin(2016x)dx
Find E.
Bonus: Generalize for ∫0π/21−sin(12nx)dx, where n is a positive integer, and prove it.
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