The basic law of the dynamics of rotational motion. Rotational movement of the body
LECTURE No. 4
BASIC LAWS OF KINETICS AND DYNAMICS
ROTATIONAL MOTION. MECHANICAL
PROPERTIES OF BIO-TISSUES. BIOMECHANICAL
PROCESSES IN THE MUSTOCULAR SYSTEM
PERSON.
1. Basic laws of kinematics of rotational motion.
Rotational movements of the body around a fixed axis are the most simple view movements. It is characterized by the fact that any points of the body describe circles, the centers of which are located on the same straight line 0 ﺍ 0 ﺍﺍ, which is called the axis of rotation (Fig. 1).
In this case, the position of the body at any time is determined by the angle of rotation φ of the radius of the vector R of any point A relative to its initial position. Its dependence on time:
(1)
is the equation of rotational motion. The speed of rotation of a body is characterized by angular velocity ω. The angular velocity of all points of the rotating body is the same. It is a vector quantity. This vector is directed along the axis of rotation and is related to the direction of rotation by the rule of the right screw:
.
(2)
When a point moves uniformly around a circle
,
(3)
where Δφ=2π is the angle corresponding to one full revolution of the body, Δt=T is the time of one full revolution, or the period of rotation. The unit of measurement of angular velocity is [ω]=c -1.
In uniform motion, the acceleration of a body is characterized by angular acceleration ε (its vector is located similar to the angular velocity vector and is directed in accordance with it during accelerated motion and in the opposite direction during slow motion):
.
(4)
The unit of measurement for angular acceleration is [ε]=c -2.
Rotational motion can also be characterized by linear speed and acceleration of its individual points. The length of the arc dS described by any point A (Fig. 1) when rotated by an angle dφ is determined by the formula: dS=Rdφ. (5)
Then the linear speed of the point :
.
(6)
Linear acceleration A:
.
(7)
2. Basic laws of the dynamics of rotational motion.
Rotation of a body around an axis is caused by a force F applied to any point of the body, acting in a plane perpendicular to the axis of rotation and directed (or having a component in this direction) perpendicular to the radius vector of the point of application (Fig. 1).
A moment of power relative to the center of rotation is a vector quantity numerically equal to the product of force
by the length of the perpendicular d, lowered from the center of rotation to the direction of the force, called the arm of the force. In Fig. 1 d=R, therefore
.
(8)
Moment rotational force is a vector quantity. Vector
applied to the center of the circle O and directed along the axis of rotation. Vector direction
consistent with the direction of force according to the right-hand screw rule. Elementary work dA i , when turning through a small angle dφ, when the body passes a small path dS, is equal to:
The measure of the inertia of a body during translational motion is mass. When a body rotates, the measure of its inertia is characterized by the moment of inertia of the body relative to the axis of rotation.
The moment of inertia I i of a material point relative to the axis of rotation is a value equal to the product of the mass of the point by the square of its distance from the axis (Fig. 2):
.
(10)
The moment of inertia of a body relative to an axis is the sum of the moments of inertia of the material points that make up the body:
.
(11)
Or in the limit (n→∞): ,
(12)
G de integration is carried out over the entire volume V. The moments of inertia of homogeneous bodies of regular geometric shape are calculated in a similar way. The moment of inertia is expressed in kg m 2.
The moment of inertia of a person relative to the vertical axis of rotation passing through the center of mass (the center of mass of a person is located in the sagittal plane slightly in front of the second cruciate vertebra), depending on the position of the person, has the following values: 1.2 kg m 2 at attention; 17 kg m 2 – in a horizontal position.
When a body rotates, its kinetic energy consists of the kinetic energies of individual points of the body:
Differentiating (14), we obtain an elementary change in kinetic energy:
.
(15)
Equating the elementary work (formula 9) of external forces to the elementary change in kinetic energy (formula 15), we obtain: , where:
or, given that
we get:
.
(16)
This equation is called the basic equation of rotational motion dynamics. This dependence is similar to Newton's II law for translational motion.
The angular momentum L i of a material point relative to the axis is a value equal to the product of the point’s momentum and its distance to the axis of rotation:
.
(17)
Momentum of impulse L of a body rotating around a fixed axis:
Angular momentum is a vector quantity oriented in the direction of the angular velocity vector.
Now let's return to the main equation (16):
,
.
Let's bring the constant value I under the differential sign and get: ,
(19)
where Mdt is called the moment impulse. If the body is not acted upon by external forces (M=0), then the change in angular momentum (dL=0) is also zero. This means that the angular momentum remains constant: .
(20)
This conclusion is called the law of conservation of angular momentum relative to the axis of rotation. It is used, for example, during rotational movements relative to a free axis in sports, for example in acrobatics, etc. Thus, a figure skater on ice, by changing the position of the body during rotation and, accordingly, the moment of inertia relative to the axis of rotation, can regulate his rotation speed.
Laboratory work No. 15
STUDYING GYROSCOPE MOTION
Goal of the work: studying the laws of rotational motion, studying the movement (precession) of a gyroscope under the influence of torque.
Theory of operation
Basic concepts. Basic law of rotational motion
Momentum of a material pointL relative to point O called vector product radius vector of this point by its momentum vector p:
Where r– radius vector drawn from point O to point A, the location of the material point, p=m v– momentum of a material point. Modulus of the angular momentum vector:
where a is the angle between the vectors r And p, l – arm of vector p relative to point O. Vector L, according to the definition of a vector product, it is perpendicular to the plane in which the vectors lie r And p(or v), its direction coincides with the direction of translational motion of the right propeller as it rotates from r to p along the shortest distance, as shown in the figure.
Momentum relative to the axis is a scalar quantity equal to the projection onto this axis of the angular momentum vector defined relative to an arbitrary point on this axis.
A moment of powerM material point relative to point O is a vector quantity determined by the vector product of the radius vector r drawn from point O to the point of application of force and force F:
. Modulus of the moment of force vector:
where a is the angle between the vectors r And F, d = r*sina – force arm – the shortest distance between the line of action of the force and point O. Vector M(as well as L) - pseudovector , it is perpendicular to the plane in which the vectors lie r And F, its direction coincides with the direction of translational movement of the right propeller as it rotates from r To F along the shortest distance, as shown in the figure. Vector value and direction M can also be calculated mathematically using the definition of a cross product.
Moment of force about the axis called a scalar quantity equal to the projection onto this axis of the vector of the moment of force M defined relative to an arbitrary point on this axis.
Basic law of the dynamics of rotational motion
To clarify the purpose of the above concepts, consider a system of two material points (particles) and then generalize the result to a system of an arbitrary number of particles (i.e., to a solid body.)
Let particles with masses m 1, m 2 be acted upon by internal f 12, f 21 and external forces F 1 And F 2.
Let us write down Newton’s second law for each of the particles, as well as the connection between internal forces arising from Newton’s third law:
Vector multiply equation (1) by r 1, and equation (2) by r 2 and add the resulting expressions:
Let us transform the left-hand sides of equation (4), taking into account that
And the vectors and are parallel and their vector product is equal to zero, then
(5
)
The first two terms on the right in (4) are equal to zero, since the internal forces f 12, f 21 equal in size and oppositely directed (vector r 1-r 2 directed along the same straight line as the vector f 12).
To derive this law, consider simplest case rotational motion of a material point. Let us decompose the force acting on a material point into two components: normal - and tangent - (Fig. 4.3). The normal component of the force will lead to the appearance of normal (centripetal) acceleration: ; , where r = OA - radius of the circle.
A tangential force will cause a tangential acceleration to appear. In accordance with Newton's second law, F t =ma t or F cos a=ma t.
Let's express the tangential acceleration in terms of the angular acceleration: a t =re. Then F cos a=mre. Let's multiply this expression by the radius r: Fr cos a=mr 2 e. Let us introduce the notation r cos a = l , Where l - leverage of force, i.e. length of the perpendicular lowered from the axis of rotation to the line of action of the force. Sincemr 2 =I - moment of inertia of a material point, and product = Fl = M - moment of force, then
Product of moment of force M for the duration of its validity dt is called the moment impulse. Product of moment of inertia I by angular velocity w is called the angular momentum of the body: L=Iw. Then the basic law of the dynamics of rotational motion in the form (4.5) can be formulated as follows: the momentum of the moment of force is equal to the change in the angular momentum of the body. In this formulation, this law is similar to Newton’s second law in the form (2.2).
End of work -
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A simpler formulation can be given the basic law of the dynamics of rotational motion(it is also called Newton's second law for rotational motion): torque is equal to the product of the moment of inertia and angular acceleration:
moment of impulse(angular momentum, angular momentum) of a body is called the product of its moment of inertia and angular velocity:
Momentum is a vector quantity. Its direction coincides with the direction of the angular velocity vector.
The change in angular momentum is determined as follows:
. (I.112)
A change in angular momentum (with a constant moment of inertia of the body) can occur only as a result of a change in angular velocity and is always due to the action of a moment of force.
According to the formula, as well as formulas (I.110) and (I.112), the change in angular momentum can be represented as:
. (I.113)
The product in formula (I.113) is called momentum impulse or driving force. It is equal to the change in angular momentum.
Formula (I.113) is valid provided that the moment of force does not change over time. If the moment of force depends on time, i.e. , That
. (I.114)
Formula (I.114) shows that: the change in angular momentum is equal to the time integral of the moment of force. In addition, if this formula is presented in the form: , then the definition will follow from it moment of force: instantaneous torque is the first derivative of angular momentum with respect to time,