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Figure shows a loop the loop track of ra...

Figure shows a loop the loop track of radius R. A car (without engine) starts from a platform at a distance h above the top of the loop and goes around the loop without falling off the track. Find the minimum value of h for a successful looping. Neglect friction.

Text Solution

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Suppose the speed of the car at the topmost point of the loop si v. Taking the gravitatioN/Al potential energy to be ero at the platform and assuming that the car starts with a negligible speed the conservation of energy shows
`0=-mgh+1/2mv^2`
or, `mv^2=2mgh`
where m is the mass of teh car. The car moving in a circle must have radial aceleration `v^2/R` at this instant. The force on the car are mg due to gravirty and N due to the contact with the tgrack. Both these forces are n radial direction at the top o the loop. Thus, from Newton's law
`mg+N=(mv^2)/R`
or, `Mg+N=2mgh/R`
For h to be minimum N should assume the minimum value which can be zero. Thus,
`2mg (h_(min))/R=mg or h_(min)=R/2`
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