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प्रश्न
Find the radius of gyration of circular ring of radius r about a line perpendicular to the plane of the ring and passing through one of its particles.
उत्तर
Moment of inertia of the ring about a point on the rim of the ring and the axis perpendicular to the plane of the ring = mR2 + mR2 = 2mR2 (from parallel axis theorem)
We know that
\[m K^2 = 2m R^2 \]
K = Radius of the gyration
\[ \Rightarrow K = \sqrt{2 R^2} = \sqrt{2}R\]
APPEARS IN
संबंधित प्रश्न
State an expression for the moment of intertia of a solid uniform disc, rotating about an axis passing through its centre, perpendicular to its plane. Hence derive an expression for the moment of inertia and radius of gyration:
i. about a tangent in the plane of the disc, and
ii. about a tangent perpendicular to the plane of the disc.
Prove the theorem of parallel axes about moment of inertia
State Brewster's law.
Prove the theorem of perpendicular axes.
(Hint: Square of the distance of a point (x, y) in the x–y plane from an axis through the origin perpendicular to the plane is x2 + y2).
Prove the theorem of parallel axes.
(Hint: If the centre of mass is chosen to be the origin `summ_ir_i = 0`
Answer in brief:
State the conditions under which the theorems of parallel axes and perpendicular axes are applicable. State the respective mathematical expressions.
A string of length ℓ fixed at one end carries a mass m at the other. The string makes 2/π revolutions/sec around the vertical axis through the fixed end. The tension in the string is ______.
State and explain the theorem of parallel axes.
A solid cylinder of radius r and mass M rolls down an inclined plane of height h. When it reaches the bottom of the plane, then its rotational kinetic energy is ____________.
(g = acceleration due to gravity)
A circular disc 'X' of radius 'R' made from iron plate of thickness 't' has moment of inertia 'Ix' about an axis passing through the centre of disc and perpendicular to its plane. Another disc 'Y' of radius '3R' made from an iron plate of thickness `("t"/3)` has moment of inertia 'Iy' about the s same as that of disc X. The relation between Ix and ly is ______.
Two bodies have their moments of inertia I and 2I respectively about their axes of rotation. If their kinetic energies of rotation are equal, their angular momenta will be in the ratio ______.
A wheel of moment of inertia 2 kg m2 is rotating at a speed of 25 rad/s. Due to friction on the axis, it comes to rest in 10 minutes. Total work done by friction is ______.
The moment of inertia of a uniform square plate about an axis perpendicular to its plane and passing through the centre is `"Ma"^2/6` where M is the mass and 'a' is the side of square plate. Moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corner is ______.
A solid sphere of mass 'M' and radius 'R' is rotating about its diameter. A disc of same mass and radius is also rotating about an axis passing through its centre and perpendicular to the plane but angular speed is twice that of the sphere. The ratio of kinetic energy of disc to that of sphere is ______.
A metal ring has a moment of inertia 2 kg·m2 about a transverse axis through its centre. It is melted and recast into a thin uniform disc of the same radius. What will be the disc's moment of inertia about its diameter?
Prove the theorem of perpendicular axes about the moment of inertia.
