a disc is set rolling on an inclined plane of angle 30 degree at its base it is in continuous rolling upto height his equal to 1m, then leave the plane . if initially it had kinetic energy 4j. find the ratio of kinetic energy die to translation and rotation at maximum height of its trajectory
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Answer:
The moment of inertia of a sphere about its central axis and a disc of mass M and radius R as shown in figure above is
I
disc
=
5
2
MR
2
Kinetic energy of the rotation of the disc, K.E
rotation
=
2
1
Iω
2
where ω is the angular velocity.
⟹K.E
rotation
=
2
1
Iω
2
=
2
1
5
2
MR
2
ω
2
Now, angular velocity, ω=
R
v
where v is the linear velocity of the sphere.
⟹K.E
rotation
=
2
1
Iω
2
=
2
1
5
2
MR
2
R
2
v
2
=
5
1
Mv
2
Kinetic energy of the linear motion K.E
linearmotion
=
2
1
Mv
2
Total kinetic energy =K.E
rotation
+K.E
linearmotion
=
5
1
Mv
2
+
2
1
Mv
2
=
10
7
Mv
2
Now, the fraction of its total energy associated with rotation =
TotalK.E
K.Eofrotation
=
10
7
Mv
2
5
1
Mv
2
=
7
2
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