Candlemass

On a camping trip a little while ago, I was watching the wood shift in the fire, and I came up with an interesting thought experiment: If you put a candle on a scale and light it, what does the scale read over time? Obviously over a long time, the measured weight will decrease, since the candle is burning away, but for a short time after lighting, would the weight appear to go up from the mass being ejected from the candle?

The candle is throwing gasses upward at some velocity, giving them momentum. This in turn pushes the candle down with the same change in momentum, similar to how a rocket works. Change in momentum is the same as a force, and we also have gravity pushing down according to the current mass:

To calculate the momentum change, I decided to use the average speed of an ideal gas:

Where k is Boltzmann's constant, T is the temperature in Kelvin, and m is the mass of the molecule. When things burn, they typically give off two types of molecules: water, and carbon dioxide. To figure out how much, we'll need to do a bit of chemistry. Candle wax is typically a large hydrocarbon, which combines with oxygen:

For beeswax, n is 45, so since I'm a physicist, I'm going to call that big and throw out those plus-2s to say that for every 1 unit of carbon burned, there are 2 units of hydrogen and 3 of oxygen. Now we can write the total change in force as 

The Wikipedia article I linked above gives a burn rate of about 0.1 g/minute. Plugging in a couple different temperatures, we can plot this function over a few minutes:

Impressively, the candle will register a greater force on the scale for around 8 minutes! Less impressively, that extra force will be around 2 thousandths of a pound. Looks like Yankee Candle won't be starting a space program anytime soon.

The Great Mubbet Caber

I may have mentioned before that I'm a big fan of the British quiz show QI. A recent episode included a discussion of an event from the Scottish Highland Games, the caber toss. In the sport, competitors must throw what amounts to a tree trunk such that it flips end-over-end and lands pointing as straight as possible away from them. You can see a video with some examples here:

I was really curious about the physics involved in determining a successful flip. I decided to make a simplified model of the sport: The competitor holds the caber at an initial angle θ, measured from the horizontal behind them. Then they run forward up to a speed v, and stop, exerting a torque on the end of the caber. It continues forward, while beginning to spin. When one end touches the ground, it sticks, and then the initial speed and gravity determine which direction it falls.

Based on the Wikipedia article I linked above, I tried a range of different lengths (5-7.75 m), angles (45-90°), and initial velocities (1-7.5 m/s or 2-17 mph). For each case, I ran the simulation and tested whether the caber fell away, or back toward the athlete. The plots below show the results, with blue indicating success, and red failure. As the length changes, the successful ranges shift, and for the shortest cabers they can even flip twice (blue region in the upper right)!

For the longest case, I also animated the throws for the 4 corners: min/max speed and angles:

As with many of these posts, I'm not sure understanding the theory would make me any more able to compete, but I'm content to use my imagination!