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babaliaris

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**PS:**By the way today I had exams in Physics and this problem was the first one I had to solve :p (unlucky) The question was to find the maximum angle θ that the pendulum can reach if we know that the magnitude of the acceleration is the same when the mass is located in the highest and the lowest point. The lowest point is when the mass passes exactly from the vertical position and the highest is when the angle θ(t) becomes maximum (which means that |V|=0).

And the problem is, if V does not change in speed, how can it become zero? How can the pendulum oscillate without the speed ever becoming zero at the maximum angle?

I will attach my solution which is exactly what I wrote in the exams today.

*The problem description goes as follows:*"An object of mass

**m**hangs from the roof with a rope of length

**L.**The maximum angle that is between the rope and the vertical line (the one that is perpendicular to the roof where the rope is attached) is Θ. The magnitude of the acceleration is the same at the highest and lowest point of the path. Find θ in radians."

My solution, in short, was the following (note that when I say Θ(t) I don't mean the maximum angle, I just mean the function over time of the angle. in other words

**Θ = max{Θ(t)}**):

Using everything I learned from this post, I found that

##a_{T} = gsin(θ(t))##

Also, I know ##a_{N} = \frac{|V|^2}{L}##

and then I found out that ##|a| = \sqrt{a_{T}^{2} + a_{N}^{2}} = \sqrt{g^{2}sin^{2}(θ(t)) + \frac{|V|^4}{L^{2}}}##

and then I said that ##|a1|= \sqrt{g^{2}sin^{2}(θ(t)) + 0} = gsin(θ(t)) ## since at the highest point the speed is zero and ##|a2| = \sqrt{g^{2}sin^{2}(θ(t)) + \frac{ |V_{max}|^4 }{L^2} }##

and if you say ##|a1| = |a2| <=>gsin(θ(t)) =\sqrt{g^{2}sin^{2}(θ(t)) + \frac{ |V_{max}|^4 }{L^2} } <=> g^{2}sin^{2}(θ(t)) = g^{2}sin^{2}(θ(t)) + \frac{|V_{max}|^4}{L^2} => V_{max} = 0## which is not correct.

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