Formulas for the speed (acceleration) of points of a rigid body, expressed through the speed (acceleration) of the pole and the angular speed (acceleration). The derivation of these formulas is from the principle that the distances between any points of the body during its movement remain constant.

Content

Basic formulas

The speed and acceleration of a point of a rigid body with a radius vector are determined by the formulas:
;
.
where is the angular velocity of rotation, is the angular acceleration. They are equal for all points of the body and can change with time t.
and - speed and acceleration of an arbitrarily selected point A with radius vector. This point is often called a pole.
Here and below, products of vectors in square brackets mean vector products.

Derivation of the formula for speed

Let us choose a rectangular fixed coordinate system Oxyz. Let's take two arbitrary points of a rigid body A and B. Let (x A , y A , z A ) And (x B , y B , z B )- coordinates of these points. When a rigid body moves, they are functions of time t. Their derivatives with respect to time t are projections of the point velocities:
, .

Let us take advantage of the fact that when a rigid body moves, the distance | AB| between points remains constant, that is, does not change with time t. Also constant is the square of the distance
.
Let's differentiate this equation with respect to time t, applying the rule for differentiating a complex function.

Let's shorten it by 2 .
(1)

Let's introduce vectors
,
.
Then the equation (1) can be represented as a scalar product of vectors:
(2) .
It follows that the vector is perpendicular to the vector. Let's use the property of a vector product. Then it can be represented as:
(3) .
where is a certain vector that we introduce only so that the condition is automatically satisfied (2) .
Let's write it down (3) as:
(4) ,

Now let's study the properties of the vector. To do this, we create an equation that does not contain the velocities of the points. Let's take three arbitrary points of a rigid body A, B and C. For each pair of these points we write the equation (4) :
;
;
.
Let's add these equations:

.
We reduce the sum of the velocities on the left and right sides. As a result, we obtain a vector equation containing only the vectors under study:
(5) .

It is easy to see that the equation (5) has a solution:
,
where is some vector that has equal value for any pairs of points on a rigid body. Then the equation (4) for the velocities of body points will take the form:
(6) .

Now Let's consider equation (5) from a mathematical point of view. If we write this vector equation in terms of components on the coordinate axes x, y, z, then the vector equation (5) is a linear system consisting of 3 equations with 9 variables:
ω BAx , ω BAy , ω BAz , ω CBx , ω CBy , ω CBz ,ω ACx , ω ACy , ω ACz .
If the equations of the system (5) are linearly independent, then their general solution contains 9 - 3 = 6 arbitrary constants. Therefore, we have not found all solutions. There are some others. To find them, we note that the solution we found completely determines the velocity vector. Therefore, additional solutions should not lead to a change in speed. Note that the vector product of two equal vectors is equal to zero. Then, if in (6) add a term proportional to the vector, then the speed will not change:


.

Then the general solution of the system (5) has the form:
;
;
,
where C BA, C CB, C AC are constants.

We'll write it out general solution of the system (5) explicitly.
ω BAx = ω x + C BA (xB - xA)
ω BAy = ω y + C BA (y B - y A)
ω BAz = ω z + C BA (zB - zA)
ω CBx = ω x + C CB (x C - x B)
ω CBy = ω y + C CB (y C - y B)
ω CBz = ω z + C CB (zC - zB)
ω ACx = ω x + C AC (x A - x C)
ω ACy = ω y + C AC (y A - y C )
ω ACz = ω z + C AC (zA - zC)
This solution contains 6 arbitrary constants:
ω x , ω y , ω z , C BA , C CB , C AC.
As it should be. Thus, we have found all terms of the general solution of the system (5) .

Physical meaning of the vector ω

As already indicated, terms of the form do not affect the values ​​of point velocities. Therefore they can be omitted. Then the velocities of the points of the rigid body are related by the relation:
(6) .

This is the angular velocity vector of the rigid body

Let's find out the physical meaning of the vector .
To do this, let us set v A = 0 . This can always be done if you choose a reference system that, at the moment in time under consideration, is moving relative to the stationary system with a speed . Let's place the origin of the reference system O at point A. Then r A = 0 . And the formula (6) will take the form:
.
The z axis of the coordinate system is directed along the vector.
According to the property of the vector product, the velocity vector is perpendicular to the vectors and . That is, it is parallel to the xy plane. Velocity vector module:
v B = ω r B sin θ = ω |HB|,
where θ is the angle between the vectors and,
|HB| is the length of the perpendicular dropped from point B to the z axis.

If the vector does not change with time, then point B moves along a circle of radius |HB| with speed
vB = |HB| ω.
That is, ω is the angular velocity of rotation of point B around point H.
Thus, we come to the conclusion that this is vector of instantaneous angular velocity of rotation of a rigid body.

Velocity of rigid body points

So, we found that the speed of an arbitrary point B of a rigid body is determined by the formula:
(6) .
It is equal to the sum of two terms. Point A is often called pole. A fixed point or a point moving at a known speed is usually chosen as a pole. The second term represents the speed of rotation of the points of the body relative to the pole A.

Since point B is an arbitrary point, then in the formula (6) you can make a substitution. Then the speed of a point of a rigid body with a radius vector is determined by the formula:
.
The speed of an arbitrary point of a rigid body is equal to the sum of the speed of translational motion of pole A and the speed of rotational motion relative to pole A.

Acceleration of rigid body points

Now let's derive a formula for the acceleration of points of a rigid body. Acceleration is the derivative of speed with respect to time. Differentiate the formula for speed
,
applying the rules for differentiating sum and product:
.
Enter the acceleration of point A
;
and angular acceleration of the body
.
Next we notice that
.
Then
.
Or
.

That is, the acceleration vector of points of a rigid body can be represented as the sum of three vectors:
,
Where
- acceleration of an arbitrarily selected point, which is often called pole;
- rotational acceleration;
- rapid acceleration.

If the angular velocity changes only in magnitude and does not change in direction, then the vectors of angular velocity and acceleration are directed along the same straight line. Then the direction rotational acceleration coincides with or opposite to the direction of the point's velocity. If the angular velocity changes in direction, then the rotational acceleration and velocity can have different directions.

Rapid acceleration always directed towards the instantaneous axis of rotation so that it intersects it at a right angle.

Let us introduce a unit vector τ associated with the moving point A and directed tangentially to the trajectory in the direction of increasing arc coordinate (Fig. 1.6). It is obvious that τ is a variable vector: it depends on l. The velocity vector v of point A is directed tangentially to the trajectory, so it can be represented as follows

where v τ =dl/dt is the projection of the vector v onto the direction of the vector τ, and v τ is an algebraic quantity. In addition, |v τ |=|v|=v.

Point acceleration

Let's differentiate (1.22) with respect to time

(1.23)

Let's transform the last term of this expression

(1.24)

Let us determine the increment of the vector τ by dl (Fig. 1.7).

As can be seen from Fig. 1.7, angle , from where , and at .

By introducing a unit vector n of the normal to the trajectory at point 1, directed towards the center of curvature, we write the last equality in vector form

Let us substitute (1.23) into (1.24) and the resulting expression into (1.22). As a result we will find

(1.26)

Here the first term is called tangential a τ , second - normal a n.

Thus, the total acceleration a of a point can be represented as the geometric sum of the tangential and normal accelerations.

Full point acceleration module

(1.27)

It is directed towards the concavity of the trajectory at an angle α to the velocity vector, and .

If the angle α is acute, then tanα>0, therefore, dv/dt>0, since v 2 /R>0 is always.

In this case, the magnitude of the speed increases over time - the movement is called accelerated(Fig. 1.8).

In the case when the speed decreases in magnitude over time, the movement is called slow(Fig. 1.9).

If the angle α=90°, tanα=∞, that is, dv/dt=0. In this case, the speed does not change in magnitude over time, and the total acceleration will be equal to the centripetal

(1.28)

In particular, the total acceleration of uniform rotational motion (R=const, v=const) is a centripetal acceleration, equal in value to a n =v 2 /R and directed all the time towards the center.

In linear motion, on the contrary, the total acceleration of the body is equal to the tangential one. In this case, a n =0, since a rectilinear trajectory can be considered a circle of infinitely large radius, and with R→∞; v 2 /R=0; a n =0; a=a τ .

Let the movement of point M be specified in a vector way, that is, the radius vector of the point is specified as a function of time

The line described by the end of a variable vector, the beginning of which is at a given fixed point, is called the hodograph of this vector. From this and from the definition of a trajectory, the rule follows: the trajectory of a point is the hodograph of its radius vector.

Let at some moment t the point occupy a position M and have a radius vector, and at that moment - a position and a radius vector (Fig. 78).

A vector connecting successive positions of a point to specified ones

moments, is called the vector of movement of a point over time. The displacement vector is expressed as follows through the values ​​of the vector function (5):

If the displacement vector is divided by the value of the gap, we obtain the vector of the average speed of the point over time

We will now reduce the gap, directing it to zero. The limit to which the average velocity vector tends as the interval decreases indefinitely is called the velocity of the point at time t or simply the velocity of point 0. In accordance with what has been said, for the velocity we obtain:

So, the velocity vector of a point is equal to the time derivative of its radius vector:

Since the secant in the limit (at ) turns into a tangent, we come to the conclusion that the velocity vector is directed tangent to the trajectory in the direction of the point’s movement.

In the general case, the speed of a point is also variable, and one can be interested in the rate of change of speed. The rate of change of velocity is called the acceleration of a point.

To determine the acceleration a, we select some fixed point A and plot the velocity vector from it at various times.

The line that the end N of the velocity vector describes is a velocity hodograph (Fig. 79). The change in the velocity vector is expressed in the fact that the geometric point N moves along the velocity hodograph, and the speed of this movement serves, by definition, as the acceleration of the point M.

1. Methods for specifying the motion of a point in a given reference system

The main tasks of point kinematics are:

1. Description of methods for specifying the movement of a point.

2. Determination of the kinematic characteristics of the movement of a point (speed, acceleration) according to a given law of motion.

Mechanical movement − change in position of one body relative to another (reference body) with which the coordinate system called reference system .

The geometric locus of successive positions of a moving point in the reference frame under consideration is called trajectory points.

Set movement − is to provide a method by which one can determine the position of a point at any time in relation to a chosen reference system. The main ways to specify the movement of a point include:

vector, coordinate and natural .

1.Vector method of specifying movement (Fig. 1).

The position of the point is determined by the radius vector drawn from the fixed point associated with the reference body: − vector equation of motion of the point.

2. Coordinate method of specifying movement (Fig. 2).

In this case, the coordinates of the point are specified as a function of time:

- equations of motion of a point in coordinate form.

These are also parametric equations for the trajectory of a moving point, in which time plays the role of a parameter. To write its equation in explicit form, we must exclude . In the case of a spatial trajectory, excluding , we obtain:

In the case of a flat trajectory

excluding , we get:

Or .

3. The natural way to define movement (Fig. 3).

In this case, set:

1) trajectory of a point,

2) the origin of the trajectory,

3) positive direction of reference,

4) law of change of arc coordinate: .

This method is convenient to use when the trajectory of a point is known in advance.

2. Speed ​​and acceleration of a point

Consider the movement of a point over a short period of time(Fig. 4):

Then is the average speed of a point over a period of time.

The speed of a point at a given time is found as the limit of the average speed at :

Point speed − this is the kinematic measure of its movement, equal to time derivative of the radius vector of this point in the reference frame under consideration.

The velocity vector is directed tangentially to the trajectory of the point in the direction of movement.

Average acceleration characterizes the change in the velocity vector over a short period of time(Fig. 5).

The acceleration of a point at a given time is found as the limit of the average acceleration at :

Point acceleration − this is a measure of the change in its speed, equal to the derivative in time from the speed of this point or the second derivative of the radius vector of the point in time .

The acceleration of a point characterizes the change in the velocity vector in magnitude and direction. The acceleration vector is directed towards the concavity of the trajectory.

3. Determination of the speed and acceleration of a point using the coordinate method of specifying motion

The connection between the vector method of specifying motion and the coordinate method is given by the relation

(Fig. 6).

From the definition of speed:

Projections of velocity on the coordinate axes are equal to the derivatives of the corresponding coordinates with respect to time

, , . .

The magnitude and direction of velocity are determined by the expressions:

The dot above here and henceforth denotes differentiation with respect to time

From the definition of acceleration:

Acceleration projections on the coordinate axes are equal to the second derivatives of the corresponding coordinates with respect to time:

, , .

The module and direction of acceleration are determined by the expressions:

, , .

4 Speed ​​and acceleration of a point using the natural method of specifying motion

4.1 Natural axes.

Determining the speed and acceleration of a point using the natural method of specifying motion

Natural axes (tangent, principal normal, binormal) are the axes of a moving rectangular coordinate system with the origin at a moving point. Their position is determined by the trajectory of movement. The tangent (with a unit vector) is directed along the tangent in the positive direction of the arc coordinate reference and is found as the limit position of the secant passing through a given point (Fig. 9). The touching plane passes through the tangent (Fig. 10), which is located as the limiting position of the plane p as point M1 tends to point M. The normal plane is perpendicular to the tangent. The line of intersection of the normal and osculating planes is the main normal. The unit vector of the main normal is directed towards the concavity of the trajectory. The binormal (with the unit vector) is directed perpendicular to the tangent and the main normal so that the vectors , and form a right-handed triple of vectors. The coordinate planes of the introduced moving coordinate system (contiguous, normal and rectifying) form a natural trihedron, which moves together with the moving point, like a rigid body. Its movement in space is determined by the trajectory and the law of change in the arc coordinate.

From the definition of point speed

where , is the unit tangent vector.

Then

, .

Algebraic speed − projection of the velocity vector onto the tangent, equal to the derivative of the arc coordinate with respect to time. If the derivative is positive, then the point moves in the positive direction of the arc coordinate.

From the definition of acceleration

− vector variable in direction and

The derivative is determined only by the type of trajectory in the vicinity of a given point, while introducing into consideration the angle of rotation of the tangent, we have , where is the unit vector of the main normal, is the curvature of the trajectory, and is the radius of curvature of the trajectory at a given point.

Let's find how the speed and acceleration of a point are calculated if the motion is given by equations (3) or (4). The question of determining the trajectory in this case has already been considered in § 37.

Formulas (8) and (10), which determine the values ​​of v and a, contain the time derivatives of the vectors . In equalities containing derivatives of vectors, the transition to dependencies between projections is carried out using the following theorem: the projection of the derivative of a vector onto an axis fixed in a given reference system is equal to the derivative of the projection of the differentiable vector onto the same axis, i.e.

1. Determining the speed of a point. Velocity vector of a point From here, based on formulas (I), taking into account that we find:

where the dot above the letter is a symbol for differentiation with respect to time. Thus, the projections of the point’s velocity onto the coordinate axes are equal to the first derivatives of the corresponding coordinates of the point with respect to time.

Knowing the projections of velocity, we will find its magnitude and direction (i.e., the angles that the vector v forms with the coordinate axes) using the formulas

2. Determination of the acceleration of a point. Acceleration vector of a point From here, based on formulas (11), we obtain:

i.e. the projections of the acceleration of a point onto the coordinate axes are equal to the first derivatives of the velocity projections or the second derivatives of the corresponding coordinates of the point with respect to time. The magnitude and direction of acceleration can be found from the formulas

where are the angles formed by the acceleration vector with the coordinate axes.

So, if the motion of a point is given in Cartesian rectangular coordinates by equations (3) or (4), then the speed of the point is determined by formulas (12) and (13), and the acceleration by formulas (14) and (15). Moreover, in the case of movement occurring in one plane, the projection onto the axis should be discarded in all formulas