Problems of the Week

Contribute a problem

2017-11-27 Advanced

         

The center of a solid sphere of radius RR is located a distance of 2R2R 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 RR with vertical separation d.d. The capacitor is charged when φ=0,\varphi=0, and is then disconnected from the voltage source. The discs are then rotated through the angle φ=90.\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 functionh(x,y)=x2144m+y2324m. 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 h0=12m.h_0 = 12\,\text{m}. What is the volume of the resulting lake ((in m3)\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=constanth(x, y) = z = \text{constant}. The volume then results from the integral V=0h0A(z)dz\displaystyle V = \int_0^{h_0} A (z)\, dz over the cross-sectional area A(z).A(z).

E=0π/21sin(2016x)dx\mathscr{E} = \displaystyle \int_{0}^{\pi/2} \sqrt{1- \sin(2016 x) }\, dx

Find E.\mathscr{E}.


Inspiration

Bonus: Generalize for 0π/21sin(12nx)dx \displaystyle \int_{0}^{\pi/2} \sqrt{1- \sin(12n x) }\, dx , where nn is a positive integer, and prove it.

×

Problem Loading...

Note Loading...

Set Loading...