Weight, Newton and the cat

Last week we thought we’d have a go at weighing our cat.  See how much weight he’s put on since we got him back in February.  As you can imagine, this isn’t an easy process for a number of reasons, but we just about managed it.  We put the kitchen scales on the floor, put a box on the kitchen scales, a couple of cat biscuits in the box (they turn the most uncooperative of felines into putty) and then finally the cat on the biscuits.



Puddy Tat remained in the box long enough to get a rough reading on the scales, but the problem was that he didn’t sit still – which meant that the reading on the scales was going up and down a lot. But why is that? Surely his weight is his weight; why does it appear to change when he moves?

This is Newton’s third law in action. When puss sits still his weight is applying a force on the ground. In return, the ground exerts an equal and opposite force on him. This force exactly balances his weight. Consequently he doesn’t move. Now, when he jumps, he actively pushes on the ground, exerting now a greater force on the ground than before. The ground reacts back on his legs, with an equal force, which is now greater than that exerted by gravity on him. So, overall, there is more force upwards on the puss (the reaction of the ground) than there is downwards (force of gravity). And up he comes (usually in the direction of somewhere he’s not supposed to be).

Now, what is it that the kitchen scales measure? It is the force that the object on top of them is exerting on them. Usually (as in the case of baking ingredients) these don’t move of their own accord, so the force measured by the scales is the same as the weight of what’s on top. But if the objects our moving around, this is no longer the case.  The scales will read either more than the weight (when puss pushes on the ground) or less (as he leaves the ground).

This is why the old question about the 60 kg circus performer, the three 2 kg weights, and the bridge that can hold 65 kg doesn’t work.  The idea is how does the guy get across the bridge carrying three 2 kg weights – his weight, plus the three weights comes to 66 kg, which is more than the bridge can take. The answer is supposed to be that he juggles them – he only ever holds two at once. But I’m afraid that doesn’t work – when the weight comes down into his hand, it exerts a force on the man (and then onto the bridge) that is greater than 2 kg – because it has to be slowed down and then accelerated back upwards. Similar arguments apply to the joke about the 2 tonnes of canaries in the back of the truck that can hold 1 tonne – they can’t all land at once. Nice joke, but physically it doesn’t hold up.

For the record, the feline weighs about 3.8 kg, plus or minus 0.1 kg. (N.B. You physics pedants may have noticed I’ve talked about weight in kg – I should of course be using Newtons for weight, but kg is the familiar unit. When I say 60 kg of weight, I mean 60 kg multiplied by the acceleration due to gravity, 9.8 metres per second squared.)

Finally, the next step, which we didn’t get to, is to close the box and ask whether poor puss is alive or dead.

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