Let radius of cylinder = r Let height of cylinder = 2h Then from geometry r2+h2 = R2 ⇒ r2 = R2−h2 Volume of cylinder V = πr2(2h) V = π(2h)(R2–h2) this volume will be maximum, when
dV
dh
= 0
dV
dh
= 2πR2−6πh2 0 = 2πR2−6πh2 R2 = 3h2 h =
R
√3
r2 = R2−(R∕√3)2 =
2
3
R2 r = √
2
3
R So, radius ‘r’ and height ‘2h’ of the cylinder having maximum possible volume is √
2
3
R and
2
√3
R respectively. Now volume of cylinder V = πr2(2h) = π(
2
3
R2).2
R
√3
=
4πR3
3√3
Mass of cylinder M' = (
M
4
3
πR3
)×
4
3√3
πR3 =
M
√3
Moment of inertia of solid cylinder about its axis I =