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I have a slight issue with the simple pendulum problem. mathematically, the equations make complete sense, but, when i look at the problem in more intuitive manner, i get the following issue. in an infinitesimal displacement, dv vector is in the theta hat direction and acceleration must be completely in the theta direction. I mean it makes sense cause in the infinitesimal limit, the path looks like a straight line. the part that I don't get is, in reality, there is a radial acceleration too. my question is how is this possible?

Qmechanic
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Ali Moradi
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  • Why do you think that acceleration must be in the direction of $\hat\theta$? – joseph h Jan 06 '22 at 06:12
  • I mean, acceleration is defined to be the vector dv to dt. So dv and acceleration must be in the same direction, right – Ali Moradi Jan 06 '22 at 06:27
  • Not in this case of radial motion. The acceleration is always perpendicular to the velocity. Please look at centripetal force in the context of a simple pendulum. – joseph h Jan 06 '22 at 06:31
  • I think i had a wrong assumption. dv is not entirely in theta hat direction. – Ali Moradi Jan 06 '22 at 07:20

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In simple words, tangential acceleration changes velocity vector amplitude (i.e. speed) and radial acceleration changes velocity vector direction.

Here is the detailed derivation for the two acceleration components in a pendulum: https://physics.stackexchange.com/a/685423/149541

In circular motion, displacement and velocity vectors are perpendicular which is obvious since velocity is tangent on the path. Your intuition is correct when it comes to tangential acceleration - it points in the direction of the velocity vector. But this acceleration component alone cannot describe how velocity vector direction changes.

Marko Gulin
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