A small block of mass m slides along the frictionless loop-the-loop track (of ra

Question

A small block of mass m slides along the frictionless loop-the-loop track (of radius r) shown in the diagram. What would the minimum height be such that the block goes all the way around the loop?

Please show work and explain every step clearly. Please don't assume I understand all the steps, because they are obvious to you. Please show where it is potential energy, where it it is kninetic etc.

THank you very much!

A small block of mass m slides along the frictionless loop-the-loop track (of radius r) shown in the diagram. What would the minimum height be such that the block goes all the way around the loop? Please show work and explain every step clearly. Please don't assume I understand all the steps, because they are obvious to you. Please show where it is potential energy, where it it is kninetic etc.

Explanation / Answer

let the body is released from height h.

PE at that height = mgh

KE at reaching the bottom of loop = PE lost = mgh .....(1)

Let the velocity at bottom be v,

m*v*v/2 = mgh --> v = sqrt(2gh)

Now in the loop at top most point, In order to complete the loop the normal force due to loop on the mass must not be zero, for critical case, it can just be equal to zero.

So writing free body equation at top most point, (assuming V be velocity at that point)

mg + N = m*V*V/r ; N = 0

So, V = sqrt(gr)

Now balancing the energy between the lowest point and top point.

PE + KE at lowest point = PE + KE at highest point

0 + mgh = m*g*2r + m*V*V/2

mgh = mgr + mgr/2 = 5mgr/2

h = 5r/2 = 2.5 r

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