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If a pendulum is experiencing SHM then:

  • $a$ is proportional to $-s$ (acceleration proportional to displacement)

  • let the proportionality constant be $\omega^2$

  • therefore $a = -\omega^2s$

  • $\omega$ represents the angular velocity of a rotating circle which paints a sine wave of the motion

My question is: can the following centripetal force formula be used to calculate the linear velocity (tangential velocity) of the pendulum which is experiencing SHM?

  • $a = v^2 / r$

or

  • $F = mv^2 / r$

where $a$ is the acceleration of SHM.

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    $\begingroup$ What is $s$ in your equations? $\endgroup$
    – BioPhysicist
    Jun 21, 2021 at 11:27
  • $\begingroup$ You need to define $v$ more precisely than "linear velocity". Velocity is a vector. $\endgroup$
    – Bill N
    Jun 21, 2021 at 12:52
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    $\begingroup$ @BillN To be fair "linear velocity" is a fairly common term that also means "tangential velocity", i.e., it is a velocity component, not a vector itself. The other part would be the "radial velocity", which in circular motion is $0$. $\endgroup$
    – BioPhysicist
    Jun 21, 2021 at 14:53

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Yes! As long as the motion is along a circle (i.e. the radius isn't changing), then the equation $F_c=mv^2/r$ will always hold.

Of course, for the pendulum both $v$ and $F_c$ vary with time, but the equation is valid at any instant in time.

However, note that you are probably mixing up accelerations here. $a=v^2/r$ is the centripetal acceleration, whereas the relevant acceleration you want to look at for SHM will be the angular acceleration.

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No, because in case of circular motion (where centripetal acceleration exists), the centripetal force or the centripetal acceleration vectors change continuously, but their magnitude remains constant.

On the other hand, in SHM, the restoring force acting on the body is directly proportional to the displacement of the body from its mean position. The mean position isn't actually the centre of the path, it is a point where the force on the body is zero, and hence, the acceleration is zero.

As acceleration varies in magnitude in SHM, you cannot use the formulae of centripetal acceleration to derive linear velocity. You need to integrate acceleration $a$ w.r.t displacement $s$ to derive the formula for velocity of the body at any given position.

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    $\begingroup$ The equation is valid, actually. The variables in $F=mv^2/r$ are just time-dependent, but the equation is still valid at each point in time. $\endgroup$
    – BioPhysicist
    Jun 21, 2021 at 11:22

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