Question 461: Centre of Mass and Rotational Motion

Question

The following bodies,(1) a ring(2) a disc(3) a solid cylinder(4) a solid sphere,of same mass 'm' and radius 'R' are allowed to roll down without slipping simultaneously from the top of the inclined plane. The body which will reach first at the bottom of the inclined plane is ___________. [Mark the body as per their respective numbering given in the question]

Accepted Answer

$$a = {{g\sin 	heta } \over {\left( {1 + {I \over {m{R^2}}}} ight)}}$$
I
R
= mR
2
, a
R
= g sin$$	heta$$/2
I
D
= $${{m{R^2}} \over 2}$$, a
D
= $${2 \over 3}$$ g sin$$	heta$$
I
SC
= $${{m{R^2}} \over 2}$$, a
SC
= $${2 \over 3}$$ g sin$$	heta$$
I
SC
= $${2 \over 5}$$mR
2
, a
SS
= $${5 \over 7}$$ g sin$$	heta$$
S = ut + $${1 \over 2}$$at
2
,
t = $$\sqrt {{{2S} \over a}} $$
$$ 	herefore $$ t $$\propto$$ $${1 \over {\sqrt a }}$$
solid sphere will take minimum time.

Centre of Mass and Rotational Motion

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