Job number №2
2. Study of the laws of motion rotary solid
Accessories: pendulum Oberbeck MTF-06 cargo caliper.
Objective: To study the rotational motion of a solid body of laws; determination of the moment of inertia of the pendulum Oberbeck experimentally and theoretically.
Brief details of the theory. The solid body can be regarded as a system of material points, the distance between them always.
The rotational motion of a solid body is called a motion in which the trajectories of all points of the body are concentric circles centered on the same straight line, called the axis of rotation.
Let the solid body rotating around a fixed axis in
the frame of reference 00 /, committed during
an infinitesimal rotation
(Figure 2.1).
The corresponding angle of rotation is
characterized by
a vector whose magnitude is equal to the
angle of rotation and the direction 00
coincides with the axis /, and so that the direction of rotation of
the
vector and meet the right-handed screw rule. value
(2.1)
characterizes
the rate of change of the angle of rotation is called the angular
velocity. vector direction coincides
with the direction of the vector. The unit of the angular velocity -
radians per second (rad / s).
(2.2)
vector module is
(2.3)
where R-radius of the circle, which describes the point under consideration. Vector pointing towards the rotation tangential to the path and is perpendicular to the plane in which lie both.
When uneven rotation value changes over time, and the time interval is incremented.
value
(2.4)
Characterizing the speed of change over time of
the angular velocity, angular acceleration is called the rotating
body. If the body is rotated about a fixed axis, the vector is
directed along this axis; in the same direction as that at the
accelerated
and rotated in the opposite direction - in slow rotation
.
The unit of angular acceleration - radians per second squared (rad /
s2).
When the rotational motion of the body change its kinematic and dynamic quantities depend on acting on the body of the rotational torque and the moment of inertia of the body.
The moment of force is
a vector quantity equal to the radius vector bringing -
,vector
drawn from O to the point of application of force A, a vector force
(2.5)
The direction of the vector perpendicular to the plane in which the force vector is the radius - vector and forms with them a pair of right-handed (Figure 2.3) is equal to the vector module
(2.6)
where R- shoulder vector with respect to the point O.
The moment of inertia
of the material point with respect to any axis is the product of its
mass in the square of the distance from the
axis
of rotation:
(2.7)
Moment of inertia of a body is equal to the sum of the moments of inertia of all the material points of the body
(2.8)
where
- an elementary mass contained in the volume - th particle. The unit
is the product of the mass
-
density of the body at a given point
on the corresponding volume element
:
(2.9)
Consequently, the moment of inertia can be written as:
(2.10)
If the density of the body is constant, it can be taken outside the sum of:
(2.11)
Relations (2.10) and (2.11) are approximate, and the more accurate the smaller the elementary volumes and the corresponding elemental mass . Therefore, the problem of finding moments of inertia reduced to the integration:
(2.12)
The moment of inertia is a physical quantity that characterizes the inertia body when changing their angular velocity under the influence of torque.
In cases where the body axis of rotation is
arbitrary, the calculation of the moment of inertia is considerably
simplified if we use the theorem of Steiner: moment of inertia
about any axis equals the sum of the moment of inertia
with respect to the geometrical axis, and the works of body weight to
the square of the distance
between the axles:
(2.13)
The Basic Law of the dynamics of rotational motion of a rigid body about a fixed axis z is written as follows:
(2.14)
Mz where - the total time of all external forces about the axis of rotation.