System of particles and Rotational motion NCERT Solutions Class 11 Physics - Solved Exercise Question 7.27

Question 7.27:
Prove the result that the velocity v of translation of a rolling body (like a ring, disc, cylinder or sphere) at the bottom of an inclined plane of a height h is given by v2 = 2gh / [1 + (k2/R2) ]
Using dynamical consideration (i.e. by consideration of forces and torques). Note k is the radius of gyration of the body about its symmetry axis, and R is the radius of the body. The body starts from rest at the top of the plane.

Solution:
A body rolling on an inclined plane of height h,is shown in the following figure:
m = Mass of the body
R = Radius of the body
K = Radius of gyration of the body
v = Translational velocity of the body
h =Height of the inclined plane
g = Acceleration due to gravity
Total energy at the top of the plane, E­1= mgh
Total energy at the bottom of the plane, Eb = KErot + KEtrans
= (1/2) I ω2 + (1/2) m v2
But I = mk2 and ω = v / R
∴ Eb = (1/2)(mk2)(v2/R2) + (1/2)mv2
= (1/2)mv2 (1 + k2 / R2)
From the law of conservation of energy, we have:
ET = Eb
mgh = (1/2)mv2 (1 + k2 / R2)
∴ v = 2gh / (1 + k2 / R2)
Hence, the given result is proved.

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