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Well, now I am completely confused after solving a sum which demands to calculate minimum velocity of a rod at the bottom most position so that it completes a vertical circle.

In case of a rod, applying energy conservation at highest and lowest points, we obtain that $\text{v}_{\text{min}} = \sqrt{4gl} \hspace{0.3cm}$ where l is the length of the rod and g is the acceleartion due to gravity.

In case of a string, the formula was $\sqrt{5gl}\hspace{0.3cm} $to complete the circle. I recall the derivation which used the fact that tension at uppermost point is zero in the limiting scenario.

Why couldn't we do the same for the string as for the rod ? As is, the work done by tension is zero.

Conversely why didn't we do the same for the rod as for the string because even the rod has tension, right ?

ps: There are two questions - rod concept for string and string concept for rod (why we can't apply)

  • NOTE : Rod is MASSLESS
  • Point particle of mass m is at the end of the rod/string
  • NO friction at center pivoted point
  • NO air drag and viscous force to be taken
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The difference between the rod and the string is that the rod can provide a compression force (the opposite of tension) but the string cannot.

In the case of the rod the particle can have a speed of zero at the top of the circle - the compression force from the rod can still support the particle’s weight $mg$.

In the case of the string, no compression force can be exerted by the string, so the minimum possible speed of the particle at the top of the circle is the speed at which its weight provides exactly the centripetal force required to keep it moving in a circle. This minimum speed is $\sqrt{gl}$.

The difference in the minimum particle speeds at the top of the circle accounts for the difference in minimum speeds at the bottom of the circle.

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The rod keeps it's shape.

The mass on the string could have enough initial speed to ensure (by energy considerations) that it reaches the top of the circle - but in the string case, it might not have completed a circle.

Near the top it might continue upwards but not along the circular path, that's why the tension is positive condition was included.

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