Exam (elaborations)
MOTION OF SYSTEM OF PARTICLES AND RIGID BODY
MOTION OF SYSTEM OF PARTICLES AND RIGID BODY
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K.V. Lumding; K.V. Karimganj; K.V. LangjingMOTION OF SYSTEM OF PARTICLES AND
RIGID BODY
CONCEPTS.
.Centre of mass of a body is a point where the entire mass of the body can be
supposed to be concentrated
For a system of n-particles, the centre of mass is given by
.Torque The turning effect of a force with respect to some axis, is called moment
of force or torque due to the force.
force from the axis of rotation.
⃗= ⃗ ⃗⃗⃗⃗
.Angular momentum (⃗⃗). It is the rotational analogue of linear momentum and is
measured as the product of linear momentum and the perpendicular distance of its
line of axis of rotation.
Mathematically: If ⃗⃗ is linear momentum of the particle and ⃗ its position vector, then
angular momentum of the particle, ⃗⃗ ⃗ ⃗⃗
(a)In Cartesian coordinates :
(b)In polar coordinates : ,
Where is angle between the linear momentum vector ⃗⃗ and the position of vector
⃗.
S.I unit of angular momentum is kg
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,K.V. Lumding; K.V. Karimganj; K.V. Langjing
Geometrically, angular momentum of a particle is equal to twice the product of
mass of the particle and areal velocity of its radius vector about the given axis.
.Relation between torque and angular momentum:
⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗
(i) ⃗ (ii) If the system consists of n-particles, then ⃗
⃗⃗
.
.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: I= =∑
SI unit of moment of inertia is kg .
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 :K= = 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=M .
.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
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, K.V. Lumding; K.V. Karimganj; K.V. Langjing
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: , where 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,
ii. M.I. of a circular ring about an axis through its centre and
perpendicular to its plane,
iii. M.I. of a circular disc about an axis through its centre and
perpendicular to its plane,
iv. M.I. of a right circular solid cylinder about its symmetry axis,
v. M.I. of a right circular hollow cylinder about its axis =
vi. M.I. of a solid sphere about its diameter,
vii. M.I. of spherical shell about its diameter,
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