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First question, why is \(U_i=mgh_f\) instead of \(U_i = mgh_i\)?
that is the total energy of the system!! the energy of the system is constant if there were no other forces acting. i.e. Kinetic Energy + Potential Energy = constant = total energy now given that the pendulum starts from rest at height h_i, K.E. = 0, total energy = P.E

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Other answers:

But with the initial height hi, isn't the PE = \(mgh_i\)? Why is it \(mgh_f\)?
|dw:1353427300243:dw|
Initial PE = mg(l-lcosθi) = mgl (1-cosθi) Final PE = mg(l-lcosθf)= mgl(1-cosθf) PE loss = change in PE = mgl[(1-cosθf) - (1-cosθi)] = mgl(cosθi - cosθf) ??
|dw:1353430286611:dw|
Yup.
since l is defined as the distanced from pivoted point to the center of mass ... the energy of the system is \( mgl (1 - \cos \theta) \) ... this is the total energy of the system ... probably experiment expects you to observe until the blob halt's to move.

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