Assuming the expression for the MI of a uniform solid sphere about its diameter, obtain the expression for its moment of inertia about a tangent.
Q 87
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State the MI of a thin rectangular plate-of mass M, length l and breadth b about its transverse axis passing through its centre. Hence find its MI about a parallel axis through the midpoint of edge of length b.
A wheel of moment of inertia $1 \mathrm{~kg} \cdot \mathrm{m}^2$ is rotating at a speed of $40 \mathrm{rad} / \mathrm{s}$. Due to the friction on the axis, the wheel comes to rest in 10 minutes. Calculate the angular momentum of the wheel, two minutes before it comes to rest.
A 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.
A circular race course track has a radius of $500 \mathrm{~m}$ and is banked at $10^{\circ}$. The coefficient of static friction between the tyres of a vehicle and the road surface is 0.25 . Compute (i) the maximum speed to avoid slipping (ii) the optimum speed to avoid wear and tear of the tyres.
A certain string $500 \mathrm{~cm}$ long breaks under a tension of $45 \mathrm{~kg}$ wt. An object of mass $100 \mathrm{~g}$ is attached to this string and whirled in a horizontal circle. Find the maximum number of revolutions that the object can make per second without breaking the string, $\left[\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^2\right.$ ]
A 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 ?
A motorcyclist is describing a circle of radius $25 \mathrm{~m}$ at a speed of $5 \mathrm{~m} / \mathrm{s}$. Find his inclination with the vertical. What is the value of the coefficient of friction between the tyres and ground ?
A 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.