A mass M slides without friction on the roller coaster track shown in figure. The curved sections of the track have radius of curvature R. The mass begins its descent from the height h . At some value of h, the mass will begin to lose contact with the track. Indicate on the diagram where the mass loses contact with the track and calculate the minimum value of h for which happens.
Text Solution
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Sol. Before the inflection point A of the track, the normal reaction of track on the mass N, is
N = 
+ mg sin θ ,
Where v is the velocity of the mass. After the inflection point,
N + 
= mg sin θ ,
For which sin θ = 
, or θ = 30º.
The mass loses contact with the track if N ≤ 0. This can only happen for the second part of the track and only if 
Th conservation of mechanical energy Mg [h – (R – R sin θ )] = 
Then requires
h – R + R sin θ ≥ 
,
or h ≥ R – 
The earliest the mass can start to lose contact with the track is at A for which θ = 30º. Hence the minimum h required is 
.
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