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Here I have a tutorial problem as follows:

The problem I have is about part a, whose answer is as follows:

When I solve the partial derivative on Vf w.r.t. r, I get Vf = mω^2rsin^2(θ)/2 +g(θ), where g(θ) is a function of θ.

However, when I take the partial derivate on Vf w.r.t. θ, I get mω^2r^2sin(θ)cos(θ)dθ/dt + dg(θ)/dθ. This is different from the centrifugal force in θ dimension and I am confused.

Here I have a tutorial problem as follows:

The problem I have is about part a, whose answer is as follows:

When I solve the partial derivative on Vf w.r.t. r, I get Vf = mω^2rsin^2(θ)/2 +g(θ), where g(θ) is a function of θ.

However, when I take the partial derivate on Vf w.r.t. θ, I get mω^2r^2sin(θ)cos(θ)dθ/dt + dg(θ)/dθ. This is different from the centrifugal force in θ dimension and I am confused.

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