Advertisements
Advertisements
Question
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.
Advertisements
Solution
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
RELATED QUESTIONS
State the theorem of perpendicular axes about moment of inertia.
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.
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)
When a 12000 joule of work is done on a flywheel, its frequency of rotation increases from 10 Hz to 20 Hz. The moment of inertia of flywheel about its axis of rotation is ______. (π2 = 10)
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 ______.
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 ______.
From a disc of radius R and mass M, a circular hole of diameter R, whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc about a perpendicular axis, passing through the centre?
A uniform cylinder of mass M and radius R is to be pulled over a step of height a (a < R) by applying a force F at its centre 'O' perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of F required is:
Two particles A and B having equal charges are placed at a distance d apart. A third charged particle placed on the perpendicular bisection of AB at distance x. The third particle experiences maximum force when ______.
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?
For a uniform rectangular sheet shown in the figure, the ratio of moments of inertia about the axes perpendicular to the sheet and passing through O (the centre of mass) and O' (corner point) is:
A thin rod of length ‘4L’ and mass ‘4m’ is bent at the points as shown in the figure. The moment of inertia of the rod about an axis passing through point ‘O’ and perpendicular to plane of the paper is:
