Dynamics of rotational motion.
§ 1 The moment of inertia. Steiner's theorem


The moment of inertia of a point
is

 

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

-integrated throughout.

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

Square ring

 

 

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


Consider a rigid body rotating around a fixed axis Z, passing through it with angular velocity
ϖ. because the body is absolutely rigid, therefore, all of the body will rotate at the same angular velocity

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


The moment of a force (torque)
about a fixed point O is called pseudovector value equal to the vector product of the radius vector  from the point O to the point of application of force, the force .

-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.
For a body of any shape and with an arbitrary distribution of mass, there are 3 mutually perpendicular passing through the center of mass of the body axis, which can serve as free axes: they are called the principal axes of inertia.

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:

Then

or

  • the basic equation of the dynamics of rotational motion.

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.
The angular momentum of a rigid body about an axis is the sum of the angular momentum of individual particles:

Differentiate with respect to dt

- the basic equation of the dynamics of rotational motion.
Generally performed vector equality

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.


§ 5 The values characterizing the translational and rotational motion, and the relationship between them:

 

Linear

Angular

Relationship

The time dependence of

Path (displacement)

 

 

 

rotation vector

 

 

 

 

linear speed

angular velocity

 

linear acceleration

angular acceleration

 

 

mass

m

moment of inertia

I

 

 

momentum,

linear momentum

 

p

angular momentum

L

 

force

F

moment of force, torque

M

 

kinetic energy

WK,

the rotational kinetic energy

WK,

 

 

elementary work

dA,

 

elementary work of the rotational motion

 

 

 

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