moment of inertia of spherical shell of radius a and mass m about a diameter
(a) 23ma 2
(b) 2/5ma 3
(d) 13 ma 3
(c) 2/7ma 4
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Answer:
Moment of Inertia of a spherical shell is
I
shell
=
3
2
mr
2
The given hollow sphere has an inner radius R and outer radius 2R.
Volume V=
3
4
π((2R)
3
−R
3
)=
3
4
π×7R
3
=
3
28
πR
3
∴ρ=
3
28
πR
3
M
Consider a thin shell of radius x and width dx for the hollow sphere.
Mass of this shell dm=ρ×(4πx
2
dx) where 4πx
2
dx is the volume of the thin shell of width dx.
∴dI=
3
2
dmx
2
=
3
2
ρ×(4πx
2
dx)x
2
∴I=∫
R
2R
dI=∫
R
2R
3
2
×ρ×(4πx
2
dx)x
2
=(
3
2
×
3
28
πR
3
M
)×(4π)∫
R
2R
x
4
dx=
7R
3
3M
(
5
1
x
5
)
∣
∣
∣
∣
∣
R
2R
⇒I=
3×7R
3
2×3M
((2R)
5
−R
5
)=
3×7R
3
2×3M
×
5
31
R
5
⇒I=
35
62
R
2
Step-by-step explanation:
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