Science and Engineering

平衡

If you're able to balance a ruler or pen on your finger, you've completed the first step on the way to a job at the legendary Chateau Apres-Ski. At this world-renowned restaurant, rather than serving trays, the wait-staff deliver orders on long, reinforced skis.

The servers are able to pull off this feat because they know that if they support the ski at itscenter of mass ( CM ) , (\text{CM}), it will be balanced. The center of mass is mathematically just the average position of all the weights. You can calculate it by adding up the mass of each object m i m_i multiplied by its position x i , x_i, 然后dividing by the total mass:

CM = x 1 m 1 + x 2 m 2 + + x n m n m 1 + m 2 + + m n . \text{CM}= \frac{x_1m_1 + x_2m_2 + \cdots + x_nm_n}{m_1 + m_2 + \cdots + m_n}.

It's simple to find the center of mass when you have two items that have the same weight: put one item on each end of the ski and support the ski at the center.

Things get a little trickier when the items have different weights or there are more than two items. We still need the ski to be supported at the average position of weight. This could be accomplished by moving your hand towards the heavier object, or by moving the heavier object away from the end of the ski.

Today's Challenge

A server at the Chateau Apres-Ski is delivering hot chocolate and a pot of cheese fondue weighing the same as 4 4 hot chocolate mugs together.

If the fondue pot is placed at the far right end, how many hot chocolates (spaced evenly from the left end of the ski to the middle) are needed so that the server can hold the ski at its center?