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Wednesday, 1 March 2017

Motion of system of particles and rigid body II


Law of conservation of angular momentum: If no external torque acts on a system, then the total angular momentum of the system always remain conserved.
Mathematically:

Moment of inertia(I): the moment of inertia of a rigid body about a given axis of rotation is the sum of the products of masses of the various particles and squares of their respective perpendicular distances from the axis of rotation.
Mathematically: 
SI unit of moment of inertia is kg m2

MI corresponding to mass of the body. However, it depends on shape & size of the body and also on position and configuration of the axis of rotation.

Radius of gyration (K): it is defined as the distance of a point from the axis of rotation at which, if whole mass of the body were concentrated, the moment of inertia of the body would be same as with the actual distribution of mass of the body.

Mathematically: 
                          
= rms distance of particles from the axis of rotation.

SI unit of gyration is m. Note that the moment of inertia of a body about a given axis is equal to the product of mass of the body and squares of its radius of gyration about that axis i.e. I = Mk2 .

Theorem of perpendicular axes: It states that the moment of inertia of a plane lamina about an axis perpendicular to its plane is equal to the sum of the moment of inertia of the lamina about any two mutually perpendicular axes in its plane and intersecting each other at the point, where the perpendicular axis passes through the lamina.
Mathematically: 

Where x & y-axes lie in the plane of the Lamina and z-axis is perpendicular to its plane and passes through the point of intersecting of x and y axes.

Theorem of parallel axes: It states that the moment of inertia of a rigid body about any axis is equal to moment of inertia of the body about a parallel axis through its center of mass plus the product of mass of the body and the square of the perpendicular distance between the axes.
Mathematically:
is moment of inertia of the body about an axis through its centre of mass and is the perpendicular distance between the two axes.

Moment of inertia of a few bodies of regular shapes:
i. M.I. of a rod about an axis through its c.m. and perpendicular to rod, I = 1/12 ML2
ii. M.I. of a circular ring about an axis through its centre and perpendicular to its plane, I = MR2
iii. M.I. of a circular disc about an axis through its centre and perpendicular to its plane,  I = 1/2 MR2
iv. M.I. of a right circular solid cylinder about its symmetry axis, I = 1/2 MR2
v. M.I. of a right circular hollow cylinder about its axis = MR2
vi. M.I. of a solid sphere about its diameter, I = 2/5 MR2
vii. M.I. of spherical shell about its diameter,  I = 2/5 MR2

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