Mechanics: Why do we approximate tanx=sinx?

[frog1944]

Hi,

When it comes to questions in mechanics for banked tracks involving a width to the track (in this case a railway), why do we do approximate sinx~tanx? E.g. rail way with a width of 1.5m and one side is 0.01m higher than other and radius 100m, find the speed a train enters the curve without experiencing any lateral force? We do the "usual" stuff and arrived at tanx=v^2 / rg -> v^2 = rgtanx, but then the solutions go that the angle "x" is small and so tanx~sinx, so they sub in v^2 = 100 x 10 x 0.01/1.5.
Why not just go tan(arcsin(0.01/1.5))? It seems a bit weird to all of a sudden start using an approximation, when it can be done without it (and arguably with very little extra effort.

Thanks

[jakesilove]

Hi,

When it comes to questions in mechanics for banked tracks involving a width to the track (in this case a railway), why do we do approximate sinx~tanx? E.g. rail way with a width of 1.5m and one side is 0.01m higher than other and radius 100m, find the speed a train enters the curve without experiencing any lateral force? We do the "usual" stuff and arrived at tanx=v^2 / rg -> v^2 = rgtanx, but then the solutions go that the angle "x" is small and so tanx~sinx, so they sub in v^2 = 100 x 10 x 0.01/1.5.
Why not just go tan(arcsin(0.01/1.5))? It seems a bit weird to all of a sudden start using an approximation, when it can be done without it (and arguably with very little extra effort.

Thanks

Hey,

Basically, this is just a common approximation that can be used in situations like this (for small angles, and when one side is much larger than the other, I think). If you want to answer the question WITHOUT the approximation, you are absolutely welcome to do so. There are times where approximating makes the problem much, much easier, and others where it does essentially nothing (which seems to be the case in this particular question). Basically, I would suggest not being too 'against' approximations, but in the 4U course, use them as little as possible (if ever. To be honest, I don't remember making those approximations in Year 12, but it's been a while!).

If you can answer a question without approximations, do. If you can't, and you can see a clever trick like the above, then do it and see where it gets you. Generally, your answers will be essentially identical.

[RuiAce]

I honestly do not understand why the answers would do that. For a 4U question that's quite unacceptable if you ask me.

Jake's pretty much explained why it works but personally I reject their solutions because of it

[frog1944]

Ok, thanks