Velocity of Simple Pendulum Formula?

Jumb0

5+ Year Member
Ok, so the velocity of pendulum bob at some angle θ made with respect to the vertical is supposed to be given by :

v = sqrt [2gL(1−cosθ)]

Which makes sense until you think of the fact that θ ought to be ZERO when the pendulum is at it's lowest point. This does not agree with the formula, however, because cos(0)=1, therefore 1-cos(0) = 0, therefore the velocity is ZERO at the bottom of the swing...but everyone knows that velocity is actually at a MAXIMUM at the bottom of the swing....

Maybe they meant that θ is the angle made by the pendulum with respect tot he HORIZONTAL...That way it would be cos(90) at the bottom, therefore velocity would be at a maximum...

Is there something simple I am overlooking here?

Cawolf

Medical Student
5+ Year Member
You seemed to have it figured out. Theta can be set however you like, but it the formula that you wrote out, assume that theta = 90 at the point where all energy is kinetic, straight down.

If you set the sum of PE and KE = 0 and then derive the velocity formula you can see how theta is described.

Jumb0

5+ Year Member
Yeah, there were two things I was not getting about this formula:

1. This formula tells you the MAXIMUM velocity ONLY. It is only good for the velocity at the very bottom.

2. The angle θ in this case refers to the angle made with the vertical IN THE INITIAL CONDITION. It is the angle that you displace the bob from equilibrium to begin the periodic motion...

It is not a general formula for the velocity of the bob at any point in its trajectory like I had erroneously assumed at first.

Cawolf

Medical Student
5+ Year Member
The formula can describe the velocity at any angle theta.

It only depends on the height of the bob, which is described by the angle.

5+ Year Member
The formula can describe the velocity at any angle theta.

It only depends on the height of the bob, which is described by the angle.
I think this formula only gives you the speed at the lowest point, i.e. the max speed. Theta in this formula is the angle of string and Y axis when the bob is at the highest position.

In order to find the speed at any point between the highest point and the lowest point (using conservation of energy), you need to take into account both the KE and PE of the bob at that point. Basically you have to assume at any point KE+PE=PE_max, where PE_max is the PE at the highest point. If you do the math, the answer will be v=sqrt (2gL(cos (theta)-cos(theta_max))), where "theta" is the angle corresponding to the desired point and "theta_max" is the angle corresponding to the bob at the highest point.

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