- #1
mmcsa
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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 sinθ 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 sinθ
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 sinθ )θ ) Any top tips would be greatly appreciated
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 sinθ 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 sinθ
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 sinθ )θ ) Any top tips would be greatly appreciated