In a \(3 \times 3 \times 3\) tic-tac-toe game, a winning line can go through each layer of the three boards (in addition to the standard winning lines in any 2D plane of a single board). Two players play on this 3-layer board with normal tic-tac-toe rules.

**Is it possible to tie in this version of tic-tac-toe?**

10 identical circles of radius 2 are externally tangential to one another, as shown.

Connecting the centers of adjacent circles gives a 10-sided polygon, which divides the circles into two areas: blue and yellow. The positive difference between the blue and yellow areas is \( k \pi.\)

What is \(k?\)

Helium gas is enclosed in a gas-tight piston which is held in position by the clamping force of an elastic spring. In mechanical equilibrium, the gas pressure \(p = -\frac FA\) corresponds to the tensioning force \(F\) of the spring divided by the area \(A\). The gas is then brought to a temperature of \(T = 600 \,\text{K}\) by rapid heating from room temperature \(T = 300 \,\text{K}\).

What is the final temperature of the gas (in units of Kelvin) after setting a new mechanical balance with the spring?

\(\)

**Details and Assumptions:**

- The volume of the gas is \(V = Ax,\) where \(A\) is the cross-sectional area and \(x\) is the piston position.
- The mechanical force of the spring equals \(F = - k x,\) where \(k\) is the spring constant.
- The piston does not move during the heating from \(300 \,\text{K}\) to \(600 \,\text{K}\).
- Helium can be treated as an ideal gas.
- Energy is transferred only by mechanical work. The piston is perfectly thermally insulated.

×

Problem Loading...

Note Loading...

Set Loading...