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A small solid sphere rolls without slipp...

A small solid sphere rolls without slipping along the track shown in figure. The sphere starts from rest from a height h above the bottom of a loop of radius R which is much larger than the radius of the sphere r. The minimum value of h for the sphere to complete the loop is:

A

2.1 R

B

2.3 R

C

2.7 R

D

2.5 R

Text Solution

Verified by Experts

The correct Answer is:
C
At the topmost point of the loop minimum value of linear speed of centre of sphere should b:
`v=sqrt(gR)` or translational kinetic energy `=K_(T) = 1/2 mv^(2) = 1/2 mgR`
In Case of pure rolling of a solid the ratio of rational to translational kinetic energy is
`K_(R)/K_(T) = 2/5` `therefore` Total kinetic energy at topmost point should be
`therefore K = (5+2)/5, K_(T) = 7/5 (1/2 mgR) = 7/10 mgR`
Now from conservation of mechanical energy: `7/10 mgR = mg(h-2R) therefore h=2,7 R`
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