The rotational analogue of force is moment of force
(also referred to as torque) If a force acts on a single
particle at a point P whose position with respect to the origin O is given by
the position vector r the moment of the force
acting on the particle with respect to the origin O is defined as the vector
product τ = r × F
The moment of force (or torque) is a vector quantity. The symbol τ stands for the Greek letter tau. The magnitude of τ is
τ = rF sin θ
the
quantity angular momentum is the rotational analogue of linear momentum. It
could also be referred to as moment of (linear) momentum
Consider a particle of mass m and linear momentum p at a position r relative to the origin O. The
angular momentum l of the particle with
respect to the origin O is defined to be
l = r × p
The
magnitude of the angular momentum vector is l = r p sin where p is the magnitude of p and θ is the angle between r and p.
A rigid body is said to be in mechanical equilibrium, if both
its linear momentum and angular momentum are not changing with time,
or equivalently, the body has neither linear acceleration nor
angular acceleration. This means
(1) the total force, i.e. the vector sum of the forces, on the
rigid body is zero;
If the total force on the body is zero, then the total linear
momentum of the body does not change with time. Eq. (7.30a) gives the
condition for the translational equilibrium of the body.
(2) The total torque, i.e. the vector sum of the torques on the
rigid body is zero, If the total torque on the rigid body is zero, the total
angular momentum of the body does not change with time. Eq. (7.30 b) gives the
condition for the rotational equilibrium of the body.
A body may be in partial equilibrium, i.e., it may be in
translational equilibrium and not in rotational equilibrium, or it may be in
rotational
equilibrium and not in translational equilibrium.
Theradius of gyration of a body about an axis
may be defined as the distance from the axis of a mass point
whose mass is equal to the mass of the whole body and whose moment of inertia
is equal to the moment of inertia of the body about the axis.
the moment of inertia of a planar body (lamina)
about an axis perpendicular to its plane is equal to the sum of its moments of
inertia about two perpendicular axes concurrent with perpendicular axis and
lying in the plane of the body.
Iz =I x +I y
The moment of inertia of a body about any axis is
equal to the sum of the moment of inertia of the body about a parallel axis
passing
through its centre of mass and the product of its
mass and the square of the distance between the two parallel axes.
Iz′ = Iz + Ma2
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