Momentum conservation again

There are some fantastic examples of momentum conservation in everyday life. This week I was attacking the leftovers from a tree removal we had a couple of weeks ago – turning the chainsaw-cut rings the tree surgeons left us into something that could be shoved into the fire come winter time (assuming no bees are in residence).

It’s very satisfying to see that log-splitter fall under gravity, hit the target smack in the centre and move smoothly through the wood sending two roughly equal pieces flying in opposite directions at similar speeds. This is a neat collision to analyze from a momentum view point. Momentum is the product of an object’s mass and its velocity – and of course has a direction associated with it. Before the collision, we have the axe head travelling downwards. After the collision, we have the axe head stationary (in the chopping block) but two pieces of wood flying horizontally towards opposite ends of the garden.

Let’s analyze this by looking at the momentum in the vertical and horizontal directions, before and after the collision. Let’s start with the horizontal. So, before the collision, there is no horizontal momentum (it’s all vertical), and after the collision there is no net horizontal momentum either – the vector sum of the momenta of the two blocks is zero, since they move in opposite directions. In fact, if the axe falls off centre and I get two unequal pieces, it’s clear that the smaller piece (lower mass) squirts sideways much faster (higher velocity) than the larger piece – again  conserving momentum.

What about vertical momentum? The head of the log-splitter clearly has momentum before the collision but doesn’t afterwards. What’s happened here? Momentum isn’t conserved in the vertical direction because there has been a vertical force acting on the log / axe system. This force is exerted on the log / axe system by the chopping block. Hit it hard with an axe, and it exerts a force back again; that’s Newton’s third law. So, when a physicist talks about momentum being conserved, what he or she means is that it is conserved in the absence of external forces. More mathematically, it can be said that momentum will be conserved when there is translational symmetry. In this system, there is horizontal translational symmetry – basically there’s nothing stopping the two halves of the block as they move sideways, but there isn’t vertical symmetry – there’s a chopping block sitting on top of the ground.

This regard for the finer points of conservation laws in physics is obviously what motivated the cat, after I had turned my back for thirty seconds, to jump on top of the chopping block to survey the situation. He’s obviously not read the proverb about curiosity and has species.

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