centripetal acceleration formula derivation...

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Abraham_Otaku
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#1
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I'll attach the file below..
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Abraham_Otaku
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what do they even mean by the sinthetha/thetha =1 as thetha tends to 0.. I think its something to do with the fact that for very small values of angles.. sinthetha=thetha, but not sure.....help.
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Callicious
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Linear expansion. You can express a sine function via the form

\sin(\theta) = \theta - \frac{1}{3!}\theta^3 + \frac{1}{5!}\theta^5 + \cdots

If the value of \theta is very nearly zero, you can see that the above expression will tend to the limit

\sin(\theta) \approx \theta
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Abraham_Otaku
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(Original post by Callicious)
Linear expansion. You can express a sine function via the form

\sin(\theta) = \theta - \frac{1}{3!}\theta^3 + \frac{1}{5!}\theta^5 + \cdots

If the value of \theta is very nearly zero, you can see that the above expression will tend to the limit

\sin(\theta) \approx \theta
ohh i see.. Thanks.
oh yeah also.. what if the angle is 0 degrees as opposed to just tending towards 0.. wouldn't that mean that in the derivation, centripetal acceleration is infinite?..
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Callicious
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(Original post by Abraham_Otaku)
ohh i see.. Thanks.
oh yeah also.. what if the angle is 0 degrees as opposed to just tending towards 0.. wouldn't that mean that in the derivation, centripetal acceleration is infinite?..
Look at your expression. If theta goes to zero, then you have a division of theta over theta- aka unity. Iirc you discuss taking limits in A-Level mathematics.
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Abraham_Otaku
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(Original post by Callicious)
Look at your expression. If theta goes to zero, then you have a division of theta over theta- aka unity. Iirc you discuss taking limits in A-Level mathematics.
Based on your reply.. I feel as if the answer is really obvious.. still don't get it though.
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Sinnoh
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(Original post by Abraham_Otaku)
ohh i see.. Thanks.
oh yeah also.. what if the angle is 0 degrees as opposed to just tending towards 0.. wouldn't that mean that in the derivation, centripetal acceleration is infinite?..
sin(x)/x is undefined if x is equal to 0, but the limit as x goes to 0 is 1 - i.e. gets arbitrarily close to 1 without being equal.
Note that undefined doesn't mean infinite either.
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Abraham_Otaku
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(Original post by Sinnoh)
sin(x)/x is undefined if x is equal to 0, but the limit as x goes to 0 is 1 - i.e. gets arbitrarily close to 1 without being equal.
Note that undefined doesn't mean infinite either.
I see. Thanks
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