State the expression for the Ml of a thin spherical shell (i.e., a thin-walled hollow sphere) about its diameter. Hence obtain the expression for its $\mathrm{Ml}$ about a tangent.
Q 90
Download our app for free and get started
Consider a uniform, thin-walled hollow sphere radius $R$ and mass M. An axis along its diameter is an axis of spherical symmetry through its centre of mass. The $\mathrm{Ml}$ of the thin spherical shell about its diameter is
$
\mathrm{I}_{\mathrm{CM}}=\frac{2}{3} M R^2
$
Let I be its $\mathrm{Ml}$ about a tangent parallel to the diameter. Here, $\mathrm{h}=\mathrm{R}=$ distance between the two axes. Then, according to the theorem of parallel axis,
$
\begin{aligned}
I & =I_{\mathrm{CM}}+M h^2 \\
& =\frac{2}{3} M R^2+M R^2=\frac{5}{3} M R^2
\end{aligned}
$
[Note : The corresponding radii of gyration are
$
k_{\mathrm{CM}}=\sqrt{\frac{I_{\mathrm{CM}}}{M}}=\sqrt{\frac{2}{3}} R \simeq 0.8165 R \text { and }
$
$
\left.k=\sqrt{\frac{1}{M}}=\sqrt{\frac{5}{3}} R \simeq 1.291 R\right]
$
$
\mathrm{I}_{\mathrm{CM}}=\frac{2}{3} M R^2
$
Let I be its $\mathrm{Ml}$ about a tangent parallel to the diameter. Here, $\mathrm{h}=\mathrm{R}=$ distance between the two axes. Then, according to the theorem of parallel axis,
$
\begin{aligned}
I & =I_{\mathrm{CM}}+M h^2 \\
& =\frac{2}{3} M R^2+M R^2=\frac{5}{3} M R^2
\end{aligned}
$
[Note : The corresponding radii of gyration are
$
k_{\mathrm{CM}}=\sqrt{\frac{I_{\mathrm{CM}}}{M}}=\sqrt{\frac{2}{3}} R \simeq 0.8165 R \text { and }
$
$
\left.k=\sqrt{\frac{1}{M}}=\sqrt{\frac{5}{3}} R \simeq 1.291 R\right]
$
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1A stone of mass $100 \mathrm{~g}$ attached to a string of length $50 \mathrm{~cm}$ is whirled in a vertical circle by giving it a velocity of $7 \mathrm{~m} / \mathrm{s}$ at the lowest point. Find the velocity at the highest point.View Solution
- 2View SolutionState an expression for the moment of inertia of a thin ring about its transverse symmetry axis.
- 3View SolutionDefine the angular momentum of a particle.
- 4A string of length $0.5 \mathrm{~m}$ carries a bob of mass $0.1 \mathrm{~kg}$ at its end. If this is to be used as a conical pendulum of period $0.4 \pi \mathrm{s}$, calculate the angle of inclination of the string with the vertical and the tension in the string.View Solution
- 5A solid sphere of mass $1 \mathrm{~kg}$ rolls on a table with linear speed $2 \mathrm{~m} / \mathrm{s}$, find its total kinetic energy.View Solution
- 6A bucket of water is tied to one end of a rope $8 \mathrm{~m}$ long and rotated about the other end in a vertical circle. Find the number of revolutions per minute such that water does not spill.View Solution
- 7A horizontal disc is rotating about a transverse axis through its centre at $100 \mathrm{rpm}$. A $20 \mathrm{gram}$ blob of wax falls on the disc and sticks to it at $5 \mathrm{~cm}$ from its axis. The moment of inertia of the disc about its axis passing through its centre is $2 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m}^2$. Calculate the new frequency of rotation of the disc.View Solution
- 8View SolutionA thin cylindrical shell of inner radius 1.5 m rotates horizontally, about a vertical axis, at an angular speed ω. A wooden block rests against the inner surface and rotates with it. If the coefficient of static friction between block and surface is 0.3, how fast must the shell be rotating if the block is not to slip and fall ?
- 9A ballet dancer spins about a vertical axis at $2.5 \pi \mathrm{rad} / \mathrm{s}$ with his arms outstretched. With the arms folded, the MI about the same axis of rotation changes by $25 \%$. Calculate the new speed of rotation in rpm.View Solution
- 10View SolutionDefine the period and frequency of revolution of a particle performing uniform circular motion (UCM) and state expressions for them. Also state their SI units.