Let's consider the simple pendulum as is displayed here or over there (page 10). The analysis of the second Newton's law in polar coordinates goes as follows:
→F=md2→rdt2,Frˆr+Fθˆθ=md2(rˆr)dt2,Frˆr+Fθˆθ=m(¨r−r˙θ2)ˆr+m(r¨θ+2˙r˙θ)ˆθ,Frˆr+Fθˆθ=marˆr+maθˆθ.
Substituing the forces we get,
−T+mgcos(θ)=mar=m(¨r−r˙θ2),−mgsin(θ)=maθ=m(r¨θ+2˙r˙θ)
Considering the restrictions r=L and ˙r=¨r=0 we get
−T+mgcos(θ)=m(−L˙θ2),−mgsin(θ)=m(L¨θ)
Answer
Yes this is the correct equation for T and yes ar≠0. In fact ar=−L˙θ2
The particle must accelerate in the normal direction in order to track a radial path. If ar=0 then the path would be a straight line.
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