At the top of the loop, the center of the circle is below the ball. So "toward center" = downward. Both gravity (mg, always down) and tension/normal force (T or N, pulling toward center = down) point downward there.
To survive the whole loop you must enter the bottom fast enough that you still have vtop,min at the top. Use energy conservation (string tension does no work — it's perpendicular to motion).
Recall What is the minimum speed at the top, and what physical condition defines it?
vtop,min=gr, defined by tension (or normal force) =0, i.e. gravity alone supplies the centripetal force.
Recall Why is the bottom speed
5gr and not gr?
Energy conservation: climbing height 2r costs 2gr in 21v2 terms (i.e. 4gr in v2). vbot2=vtop2+4gr=gr+4gr=5gr.
Recall For a rigid rod instead of a string, what changes?
The rod can push, so T may be negative; the minimum-speed condition becomes vtop≥0, giving vbot,min=2gr.
Recall (Feynman) Explain to a 12-year-old why slow-at-the-top makes the ball fall.
Imagine swinging a ball on a string fast over your head. The string stays tight because the ball wants to fly straight but the string yanks it into a circle. If you swing too slowly at the very top, gravity is already pulling the ball down harder than it needs to curve — so the string goes floppy and the ball just drops instead of circling. The "just fast enough" speed is when gravity pulls exactly as hard as the circle needs.
Socho ek ball ko string se vertical circle me ghuma rahe ho. Sabse important point hai top — wahan circle ka center neeche hota hai, isliye "center ki taraf" ka matlab hai neeche. Gravity (mg) bhi neeche, aur string ka tension bhi neeche (center ki taraf khinch raha hai). Dono add hote hain. Centripetal requirement kehti hai net inward force =rmv2 hona chahiye. Agar ball slow ho jaye, to required force kam ho jata hai, isliye tension bhi kam hota hai. Jab tension exactly 0 ho jaye, tabhi minimum speed milti hai: vtop=gr. Isse slow chalaoge to string dheeli (slack) ho jayegi aur ball circle chhod degi.
Ab pura loop complete karne ke liye bottom par kitni speed chahiye? Yahan energy conservation lagao, kyunki tension koi kaam nahi karta (motion ke perpendicular hota hai). Bottom se top tak height 2r chढ़ni padti hai, to KE kam hoti hai. Isse vbot2=vtop2+4gr=gr+4gr=5gr, yani vbot=5gr. Ye yaad rakhne ka easy trick: bottom-side-top = 5-3-1 (sab gr ke units me, squared speeds).
Ek common galti: log sochte hain top par tension aur gravity opposite directions me hain. Nahi! Top par dono neeche point karte hain kyunki center neeche hai. Hamesha pehle poocho — center kahan hai? Doosri galti: bottom par minimum speed gr maan lena — galat, woh top ki condition hai, bottom par 5gr chahiye.
Aur ek interesting baat: agar string ki jagah rigid rod ho, to rod push bhi kar sakta hai, isliye top par tension negative ho sakta hai — tab minimum speed condition vtop=0 ban jati hai. Matlab gr wala rule sirf string ya track ke liye hai, universal nahi. Yahi physics ka maza hai — formula yaad rakhne se zyada important hai samajhna ki woh kab apply hota hai.
