Dynamics of rotational motion.
The moment of inertia of the system with respect to the axis of rotation is called a physical quantity that is equal to the sum of the product of the masses n material points of the squares of their distances from the axis of.
The moment of inertia of the body in the event of a continuous distribution of mass is
1. We find the moment of inertia of a uniform disk about an axis perpendicular to the plane of the disk and through its center. We divide the disk into annular layers of thickness dr. All points of the layer will be the same distance from the axis equal to r. The volume of such a layer is
2. Walled hollow cylinder of radius R (a hoop, a bicycle wheel, and the like).
3. Solid cylinder or disk of radius R
4. Direct thin long rod axis is perpendicular to the rod and passing through its middle.
5. Ball of radius R, about an axis passing through the center
If you know the moment of inertia about an axis passing through its center of mass, moment of inertia about any axis parallel to this, is determined with the help of Theorem Steiner: the moment of inertia with respect to the I axis of rotation is parallel to the moment of inertia ≤C relatively parallel to the axis passing through the center of mass C body, with a folded piece of body mass m and the square of the distance between the axes of the
6. The moment of inertia of a straight rod length, the axis perpendicular to the rod and passing through its end.
§ 2 The kinetic energy of rotation
If we break the body in small amounts to the elementary masses m1, m2 ... at a distance r1, r2 ..., from the axis of rotation, the kinetic energy of the body can be written as
It is known that or then
From a comparison of WK rot with Wk translational motion that the moment of inertia of the rotary motion replaces mass in the rotational motion and is a measure of the inertia of the body.
If the body is involved in translational and rotational motions at the same time, it
For example, a cylinder rolling without slipping on a plane.
§ 3 Torque.
The dynamic equation of rotational motion of a rigid body
-pseudovector its direction coincides with the plane of motion of the right screw as it rotates from to . The direction of the torque can also be defined by the rule of his left hand, four fingers of his left hand to put in the direction of the first factor , the second factor is in the palm, bent at right angles to the thumb indicates the direction of the torque . Moment of the force vector is always perpendicular to the plane in which the vectors and
- where the shortest distance between the line of action of the force and the point 0-called shoulder strength.
-strength shoulder on force
The moment of a force about a fixed axis Z is called a scalar quantity equal to the projection on the axis of the torque defined relative to an arbitrary point O of the axis Z. If the Z-axis is perpendicular to the plane in which the vectors and , ie coincides with the direction , then a moment of force represented as a vector coincides with the axis
Axis, whose position in space remains unchanged during the rotation around the body in the absence of external forces, called the free axis of the body.
We find an expression for rotational motion of the body. Let the mass m solid external force . Then the work of this force for the time dt is
Feasible in the mixed product of vectors a cyclic permutation of the factors
Work rotation of the body is the product of the moment of the force on the angle of rotation of the body dφ. Work is to increase its kinetic energy:
If the axis of rotation coincides with the main axis of inertia through the center of mass, then the vector equality
I - principal moment of inertia (moment of inertia with respect to the main axis)
§ 4 The angular momentum. The law of conservation of angular momentum
Angular momentum of a particle A relative to a fixed point 0 is a physical quantity, defined by the vector product
- radius vector from the point O to the point A
- the momentum of a particle.
- pseudovector, its direction is determined by the left-hand rule.
Angular momentum of a rigid body about a fixed axis Z is called a scalar quantity equal to the projection on the axis of the angular momentum, defined relative to an arbitrary point O that axis. The value of the angular momentum Lz is independent of the point O on the axis Z.
Differentiate with respect to dt
- the basic equation of the dynamics of rotational motion.
In a closed system the moment of the external forces is zero
The law of conservation of angular momentum: the angular momentum of a closed system is conserved, ie does not change over time.