Determining linear velocity of pendulum

[mmcsa]

Hello,

I'm trying to develop a pendulum to test protective equipment so I want to work out the length I'll need to generate a desired velocity and the necessary mass I'll need for a specific moment of inertia. I know there are multiple ways to solve for linear velocity with equating Ek and Ep being the most common. I have also tried to do this using angular acceleration, given θ ̈ = - g/l sin⁡θ however they both seem to give me different velocities, so I think I've gone wrong somewhere!

So I have 2 methods which should mathematically be the same but I can't seem to match them

Method 1: Ek = Ep
Ep=Ek
mgh = 1/2 mv^2
v=√2gh
h = L - (Lcos⁡(-θ))
v_bottom= √(2gL(1-cos⁡(θ)))
http://www.sparknotes.com/testprep/books/sat2/physics/chapter8section5.rhtml

Method 2: equations of angular motion

angular acceleration= θ ̈ = - g/l sin⁡θ
https://en.wikipedia.org/wiki/Inverted_pendulum#Stationary_pivot_point
ω^2= ω_o^2+2θ ̈ θ
V_tangential=ωr
r=radius of rotation=L for pendulum
V_tangential=L√(ω_o^2+2(-g/L sin⁡θ )θ ) Any top tips would be greatly appreciated

 

[jtbell]

Method 2: equations of angular motion

angular acceleration= θ ̈ = - g/l sin⁡θ

This gives the angular acceleration when the pendulum angle is θ. As the angle varies (as the pendulum "falls"), so does the angular acceleration.

ω^2= ω_o^2+2θ ̈ θ

This assumes the angular acceleration is constant, which is not true for your situation.

 

[mmcsa]

Thanks jtbell,
Is there an alternative equation of motion I can use to work out the velocity at each point given the constantly changing angular acceleration of the system? Or is it possible with method 1 to find the resultant velocity for other cases not just at the bottom of the pendulum swing? For instance I am considering using an inverted pendulum style design where the mass is released at say 179° to vertical and it impacts at horizontal 90° (Vy=Vresultant) therefore the calculated h would be ~ pendulum rod length

 

[Nidum]

Google ' Charpy Pendulum ' and ' Charpy impact testing '

 

[jtbell]

Or is it possible with method 1 to find the resultant velocity for other cases not just at the bottom of the pendulum swing?

Yes, you can use conservation of energy for such situations: $$E_{k,final} + E_{p,final} = E_{k,initial} + E_{p,initial} \\ \frac 1 2 mv^2_{final} + mgh_{final} = \frac 1 2 mv^2_{initial} + mgh_{initial}$$ where presumably $v_{initial} = 0$. You find the two h's from the corresponding angles by using some geometry. Of course m cancels out. This assumes that the mass of the pendulum string or rod is negligible compared to the mass of the bob.