velocity of the pendulum bob at the equilibrium position

In this post, we will derive an equation for the velocity of the pendulum bob at the equilibrium position.

Calculating the velocity of the pendulum bob at the equilibrium position requires that we arrange our coordinate system so that the height of the bob at the equilibrium position is zero.

Then the total mechanical energy is equal to the kinetic energy at the equilibrium point where Potential energy (U) = 0.

The total mechanical energy is also equal to the total potential energy at θmax where KE = 0.

Putting these conditions together, we get

E = (1/2)mV2=mgh …………………. (1)

Now, we have to find out the expression of h. For this, we will refer to figure 1.

diagram of a pendulum to find out the velocity of the pendulum bob at the equilibrium position
figure 1: pendulum

From the figure, we see that h=L – L cos θmax ………….. (2)

If we plug that value into equation 1 above, we can solve for v:

(1/2)mv2 = mg (L – L cos θmax )

v = [2gL(1-cos θmax)]1/2

velocity formula or equation of the pendulum bob at the equilibrium position
velocity equation of the pendulum bob at the equilibrium position

What we get from this equation of pendulum bob velocity at the equilibrium is:

  • The longer the string the faster the pendulum bob will move.
  • The greater the angle, the faster the pendulum bob will move.
velocity of the pendulum bob at the equilibrium position
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