Center of gravity of a rigid body
Center of gravity of a solid body is a geometric point that is rigidly connected to this body and is the center of parallel gravitational forces applied to individual elementary particles of the body (Figure 1.6).
Radius vector of this point
Figure 1.6
For a homogeneous body, the position of the center of gravity of the body does not depend on the material, but is determined by the geometric shape of the body.
If the specific gravity of a homogeneous body γ , weight of an elementary particle of a body
P k = γΔV k (P = γV)
substitute into the formula to determine rC , we have
From where, projecting onto the axes and passing to the limit, we obtain the coordinates of the center of gravity of a homogeneous volume
Similarly for the coordinates of the center of gravity of a homogeneous surface with area S (Figure 1.7, a)
Figure 1.7
For the coordinates of the center of gravity of a homogeneous line of length L (Figure 1.7, b)
Methods for determining the coordinates of the center of gravity
Based on the general formulas obtained earlier, we can indicate methods for determining the coordinates of the centers of gravity of solid bodies:
Figure 1.8
Figure 1.9
11. Basic concepts of kinematics. Kinematics of a point. Methods for specifying the movement of a point. Speed and acceleration of a point.
Basic concepts of kinematics
Kinematics- a branch of mechanics that studies the movement of bodies without taking into account the reasons that caused this movement.
The main task of kinematics is to find the position of a body at any time if its position, speed and acceleration at the initial time are known.
Mechanical movement- this is a change in the position of bodies (or parts of the body) relative to each other in space over time.
To describe mechanical motion, it is necessary to choose a reference system.
Reference body- a body (or group of bodies), taken in this case as motionless, relative to which the movement of other bodies is considered.
Reference system- this is the coordinate system associated with the reference body, and the chosen method of measuring time (Fig. 1).
The position of the body can be determined using the radius vector r⃗ r→ or using coordinates.
Radius vector r⃗ r→ points Μ - a directed straight line segment connecting the origin ABOUT with a dot Μ (Fig. 2).
Coordinate x points Μ is the projection of the end of the radius vector of the point Μ per axis Oh. Usually a rectangular coordinate system is used. In this case, the position of the point Μ on a line, plane and in space are determined, respectively, by one ( x), two ( X, at) and three ( X, at, z) numbers - coordinates (Fig. 3).
In an elementary course, physicists study the kinematics of the motion of a material point.
Material point- a body whose dimensions can be neglected under given conditions.
This model is used in cases where the linear dimensions of the bodies under consideration are much smaller than all other distances in a given problem or when the body moves translationally.
Progressive is the movement of a body in which a straight line passing through any two points of the body moves while remaining parallel to itself. During translational motion, all points of the body describe the same trajectories and at any moment of time have the same speeds and accelerations. Therefore, to describe such a movement of a body, it is enough to describe the movement of one arbitrary point.
In what follows, the word “body” will be understood as a “material point”.
The line that a moving body describes in a certain frame of reference is called trajectory. In practice, the shape of the trajectory is specified using mathematical formulas ( y = f(x) - trajectory equation) or depicted in the figure. The type of trajectory depends on the choice of reference system. For example, the trajectory of a body freely falling in a carriage that moves uniformly and rectilinearly is a straight vertical line in the frame of reference associated with the carriage, and a parabola in the frame of reference associated with the Earth.
Depending on the type of trajectory, rectilinear and curvilinear movement are distinguished.
Path s- a scalar physical quantity determined by the length of the trajectory described by the body over a certain period of time. The path is always positive: s > 0.
MovingΔr⃗ Δr→ of a body for a certain period of time - a directed straight line segment connecting the initial (point M 0) and final (dot M) body position (see Fig. 2):
Δr⃗ =r⃗ −r⃗ 0, Δr→=r→−r→0,
where r⃗ r→ and r⃗ 0 r→0 are the radius vectors of the body at these moments of time.
Projection of movement onto the axis Ox
Δrx=Δx=x−x0 Δrx=Δx=x−x0
Where x 0 and x- coordinates of the body at the initial and final moments of time.
The travel module cannot be larger than the path
|Δr⃗ |≤s |Δr→|≤s
The equal sign refers to the case of rectilinear motion, if the direction of motion does not change.
Knowing the displacement and initial position of the body, you can find its position at time t:
r⃗ =r⃗ 0+Δr⃗ ; r→=r→0+Δr→;
(x=x0+Δrx;y=y0+Δry. (x=x0+Δrx;y=y0+Δry.
Speed
The average speed hυ⃗ i hυ→i is a vector physical quantity, numerically equal to the ratio of the movement to the period of time during which it occurred, and directed along the movement (Fig. 4):
hυ⃗ i=Δr⃗ Δt;hυ⃗ i⇈Δr⃗ . hυ→i=Δr→Δt;hυ→i⇈Δr→.
The SI unit of speed is meter per second (m/s).
The average speed found using this formula characterizes the movement only on that section of the trajectory for which it is determined. On another part of the trajectory it may be different.
Sometimes they use average speed
hυi=sΔt hυi=sΔt
Where s is the path traveled over a period of time Δ t. The average speed of a path is a scalar quantity.
Instantaneous speedυ⃗ υ→ of the body - the speed of the body at a given moment of time (or at a given point of the trajectory). It is equal to the limit to which the average speed tends over an infinitesimal period of time υ⃗ =limΔt→0Δr⃗ Δt=r⃗ ′ υ→=limΔt→0Δr→Δt=r→ ′. Here r⃗ ′ r→ ′ is the derivative of the radius vector with respect to time.
In projection onto the axis Oh:
υx=limΔt→0ΔxΔt=x′. υx=limΔt→0ΔxΔt=x′.
The instantaneous speed of the body is directed tangentially to the trajectory at each point in the direction of movement (see Fig. 4).
Acceleration
Average acceleration- a physical quantity numerically equal to the ratio of the change in speed to the time during which it occurred:
ha⃗ i=Δυ⃗ Δt=υ⃗ −υ⃗ 0Δt. ha→i=Δυ→Δt=υ→−υ→0Δt.
The vector ha⃗ i ha→i is directed parallel to the velocity change vector Δυ⃗ Δυ→ (ha⃗ i⇈Δυ⃗ ha→i⇈Δυ→) towards the concavity of the trajectory (Fig. 5).
Instant acceleration:
a⃗ =limΔt→0Δυ⃗ Δt=υ⃗ ′. a→=limΔt→0Δυ→Δt=υ→ ′.
The SI unit of acceleration is meter per second squared (m/s2).
In general, instantaneous acceleration is directed at an angle to the velocity. Knowing the trajectory, you can determine the direction of speed, but not acceleration. The direction of acceleration is determined by the direction of the resultant forces acting on the body.
In rectilinear motion with increasing speed (Fig. 6, a), the vectors a⃗ a→ and υ⃗ 0 υ→0 are codirectional (a⃗ ⇈υ⃗ 0 a→⇈υ→0) and the projection of acceleration on the direction of movement is positive.
In rectilinear motion with a decreasing velocity (Fig. 6, b), the directions of the vectors a⃗ a→ and υ⃗ 0 υ→0 are opposite (a⃗ ↓υ⃗ 0 a→↓υ→0) and the projection of acceleration onto the direction of motion is negative.
Vector a⃗ a→ during curvilinear motion can be decomposed into two components directed along the speed a⃗ τ a→τ and perpendicular to the speed a⃗ n a→n (Fig. 1.7), a⃗ τ a→τ is the tangential acceleration, characterizing the speed of change in the velocity modulus during curvilinear motion, a⃗ n a→n - normal acceleration, characterizing the speed of change in the direction of the velocity vector during curvilinear motion Acceleration modulus a=a2τ+a2n−−−−−−√ a=aτ2+an2.
Methods for specifying point movement
To specify the movement of a point, you can use one of the following three methods:
1) vector, 2) coordinate, 3) natural.
1. Vector method of specifying the movement of a point.
Let the point M moves with respect to some frame of reference Oxyz. The position of this point at any time can be determined by specifying its radius vector drawn from the origin ABOUT exactly M(Fig. 3).
Fig.3
When the point moves M the vector will change over time both in magnitude and direction. Therefore, it is a variable vector (function vector) depending on the argument t:
Equality defines the law of motion of a point in vector form, since it allows us to construct a corresponding vector at any time and find the position of the moving point.
The geometric location of the ends of the vector, i.e. hodograph this vector determines the trajectory of the moving point.
2. Coordinate method of specifying the movement of a point.
The position of a point can be directly determined by its Cartesian coordinates x, y, z(Fig. 3), which will change over time as the point moves. To know the law of motion of a point, i.e. its position in space at any moment in time, you need to know the coordinates of the point for each moment in time, i.e. know dependencies
x=f 1 (t), y=f 2 (t), z=f 3 (t).
The equations are the equations of motion of a point in rectangular Cartesian coordinates. They determine the law of motion of a point using the coordinate method of specifying motion.
To obtain the trajectory equation, it is necessary to exclude the parameter t from the equations of motion.
It is not difficult to establish a relationship between the vector and coordinate methods of specifying motion.
Let's decompose the vector into components along the coordinate axes:
where r x , r y , r z - projections of the vector on the axis; – unit vectors directed along the axes, unit vectors of the axes.
Since the origin of the vector is at the origin of coordinates, the projections of the vector will be equal to the coordinates of the point M. That's why
If the point's movement is specified in polar coordinates
r=r(t), φ = φ(t),
where r is the polar radius, φ is the angle between the polar axis and the polar radius, then these equations express the equation of the trajectory of a point. Eliminating the parameter t, we get
r = r(φ).
Example 1. The motion of a point is given by the equations
Fig.4
To exclude time, the parameter t, we find from the first equation sin2t=x/2, from the second cos2t=y/3. Then square it and add it. Since sin 2 2t+cos 2 2t=1, we get . This is the equation of an ellipse with semi-axes 2 cm and 3 cm (Fig. 4).
Start point position M 0 (at t=0) is determined by the coordinates x 0 =0, y 0 =3 cm.
After 1 sec. the point will be in position M 1 with coordinates
x 1 =2sin2=2∙0.91=1.82 cm, y 1 =2cos2=3∙(-0.42)= -1.25 cm.
Note.
The movement of a point can be specified using other coordinates. For example, cylindrical or spherical. Among them there will be not only linear dimensions, but also angles. If necessary, you can familiarize yourself with specifying motion using cylindrical and spherical coordinates from textbooks.
3. A natural way to specify the movement of a point.
Fig.5
The natural way of specifying movement is convenient to use in cases where the trajectory of a moving point is known in advance. Let the curve AB is the trajectory of the point M when it moves relative to the reference system Oxyz(Fig. 5) Let’s choose some fixed point on this trajectory ABOUT", which we take as the origin of reference, and set positive and negative reference directions on the trajectory (as on the coordinate axis).
Then the position of the point M on the trajectory will be uniquely determined by the curvilinear coordinate s, which is equal to the distance from the point ABOUT' to the point M, measured along the arc of the trajectory and taken with the appropriate sign. When moving point M moves to positions M 1 , M 2 ,... . therefore the distance s will change over time.
To know the position of a point M on the trajectory at any time, you need to know the dependence
The equation expresses the law of motion of a point M along the trajectory. The function s= f(t) must be unique, continuous and differentiable.
The positive direction of reference of the arc coordinate s is taken to be the direction of motion of the point at the moment when it occupies position O. It should be remembered that the equation s=f(t) does not determine the law of motion of the point in space, since to determine the position of the point in space you need to know the trajectory of a point with the initial position of the point on it and a fixed positive direction. Thus, the movement of a point is considered given in a natural way if the trajectory and the equation (or law) of the point’s movement along the trajectory are known.
It is important to note that the arc coordinate of the point s is different from the path σ traversed by the point along the trajectory. During its movement, a point passes a certain path σ, which is a function of time t. However, the distance traveled σ coincides with the distance s only when the function s = f(t) changes monotonically with time, i.e. when a point moves in one direction. Let's assume that point M moves from M 1 to M 2. The position of the point in M 1 corresponds to time t 1, and the position of the point in M 2 corresponds to time t 2. Let us decompose the time interval t 2 - t 1 into very small time intervals ∆t 1 (i = 1.2, ...n) so that in each of them the point moves in one direction. Let us denote the corresponding increment of the arc coordinate as ∆s i . The path σ traversed by the point will be a positive value:
If the movement of a point is specified by the coordinate method, then the path traveled is determined by the formula
where dx=xdt, dy= ydt, dz=zdt.
Hence,
Example 2. The point moves in a straight line, according to the law s=2t+3 (cm) (Fig. 6).
Fig.6
At the beginning of the movement, at t=0 s=OM 0 =s 0 =3 cm. Position of the point M 0 is called starting position. At t=1 s, s=OM 1 =5 cm.
Of course, in 1 second. the point has traveled the distance M 0 M 1 = 2cm. So s– this is not the path traveled by the point, but the distance from the origin to the point.
Point velocity vector
One of the main kinematic characteristics of the movement of a point is a vector quantity called the speed of the point. The concept of the speed of a point in uniform rectilinear motion is one of the elementary concepts.
Speed- a measure of the mechanical condition of the body. It characterizes the speed of change of body position relative to a given reference system and is a vector physical quantity.
The unit of speed is m/s. Other units are often used, for example, km/h: 1 km/h=1/3.6 m/s.
The movement of a point is called uniform if the increments of the radius vector of the point over equal periods of time are equal to each other. If the trajectory of the point is a straight line, then the motion of the point is called rectilinear.
For uniformly linear motion
∆r= v∆t, (1)
Where v– constant vector.
Vector v called the speed of rectilinear and uniform motion completely determines it.
From relation (1) it is clear that the speed of rectilinear and uniform motion is a physical quantity that determines the movement of a point per unit time. From (1) we have
Vector direction v indicated in Fig. 6.1.
Fig.6.1
For uneven movement, this formula is not suitable. Let us first introduce the concept of the average speed of a point over some period of time.
Let the moving point be at the moment of time t pregnant M, determined by the radius vector, and at the moment t 1 comes to the position M 1 defined by a vector (Fig. 7). Then the movement of the point over the period of time ∆t=t 1 -t is determined by a vector which we will call the vector of movement of the point. From the triangle OMM 1 it is clear that ; hence,
Rice. 7
The ratio of the vector of movement of a point to the corresponding period of time gives a vector quantity called the average velocity of the point in absolute value and direction over the period of time ∆t:
The speed of a point at a given time t is the vector quantity v to which the average speed v cf tends as the time interval ∆t tends to zero:
So, the velocity vector of a point at a given time is equal to the first derivative of the radius vector of the point with respect to time.
Since the limiting direction of the secant MM 1 is a tangent, then the point’s velocity vector at a given time is directed tangent to the point’s trajectory in the direction of movement.
Determining the speed of a point using the coordinate method of specifying movement
The point velocity vector, taking into account that r x =x, r y =y, r z =z, we find:
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 α, β, γ that the vector v forms with the coordinate axes) using the formulas
So, the numerical value of the speed of a point at a given time is equal to the first derivative of the distance (curvilinear coordinate) s points in time.
The velocity vector is directed tangentially to the trajectory, which is known to us in advance.
Determining the speed of a point using the natural method of specifying movement
The speed value can be defined as the limit (∆r – chord length MM 1):
where ∆s – arc length MM 1 . The first limit is equal to unity, the second limit is the derivative ds/dt.
Consequently, the speed of a point is the first time derivative of the law of motion:
The velocity vector is directed, as was established earlier, tangent to the trajectory. If the velocity value at a given moment is greater than zero, then the velocity vector is directed in a positive direction
Point acceleration vector
Acceleration- vector physical quantity characterizing the rate of change of speed. It shows how much the speed of a body changes per unit time.
The SI unit of acceleration is meter per second squared. to the corresponding time interval ∆t determines the vector of the average acceleration of the point over this period of time:
The average acceleration vector has the same direction as the vector, i.e. directed towards the concavity of the trajectory.
Acceleration of a point at a given time t is called a vector quantity to which the average acceleration tends as the time interval ∆t tends to zero: The acceleration vector of a point at a given time is equal to the first derivative of the velocity vector or the second derivative of the radius vector of the point with respect to time.
The acceleration of a point is zero only when the speed of the point v constant both in magnitude and direction: this corresponds only to rectilinear and uniform motion.
Let's find how the vector is located in relation to the trajectory of the point. In rectilinear motion, the vector is directed along the straight line along which the point moves. directed towards the concavity of the trajectory and lies in the plane passing through the tangent to the trajectory at the point M and a line parallel to the tangent at an adjacent point M 1 (Fig. 8). In the limit when the point M strives for M, this plane occupies the position of the so-called osculating plane, i.e. the plane in which an infinitesimal rotation of the tangent to the trajectory occurs during an elementary movement of a moving point. Therefore, in the general case, the acceleration vector lies in the contacting plane and is directed towards the concavity of the curve.
Determination of acceleration using the coordinate method of specifying motion
The acceleration vector of a point in projection on the axis is obtained:
those. the projection of the acceleration of a point onto the coordinate axes is 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
Fig.10
Acceleration projections a x = =0, a y = =-8 cm∙s -2. Since the projection of the acceleration vector onto the axis x is equal to zero, and on the axis y– is negative, then the acceleration vector is directed vertically downward, and its value is constant and does not depend on time.
Any body can be considered as a collection of material points, which can, for example, be taken as molecules. Let the body consist of n material points with masses m1, m2, ...mn.
Center of mass of the body, consisting of n material points is called a point (in a geometric sense), the radius vector of which is determined by the formula:
Here R1 is the radius vector of point number i (i = 1, 2, ... n).
This definition looks unusual, but in fact it gives the position of the very center of mass, about which we have an intuitive idea. For example, the center of mass of the rod will be in its middle. The sum of the masses of all points included in the denominator of the above formula is called the mass of the body. Body weight called the sum of the masses of all its points: m = m1 + m2 + ... + mn.
In symmetrical homogeneous bodies, the CM is always located at the center of symmetry or lies on the axis of symmetry if the figure does not have a center of symmetry. The center of mass can be located both inside the body (disc, square, triangle) and outside it (ring, frame, square).
For a person, the position of the COM depends on the posture adopted. In many sports, an important component of success is the ability to maintain balance. So, in gymnastics, acrobatics
a large number of elements will include different types of equilibrium. The ability to maintain balance in figure skating and speed skating, where the support has a very small area, is important.
The conditions for equilibrium of a body at rest are the simultaneous equality to zero of the sum of forces and the sum of moments of forces acting on the body.
Let's find out what position the axis of rotation should occupy so that the body fixed to it remains in balance under the influence of gravity. To do this, let's break the body into many small pieces and draw the gravity forces acting on them.
In accordance with the rule of moments, for equilibrium it is necessary that the sum of the moments of all these forces about the axis equals zero.
It can be shown that for each body there is a single point where the sum of the moments of gravity about any axis passing through this point is equal to zero. This point is called the center of gravity (usually coincides with the center of mass).
Body center of gravity (CG) called the point relative to which the sum of the moments of gravity acting on all particles of the body is equal to zero.
Thus, gravity forces do not cause the body to rotate around the center of gravity. Therefore, all gravitational forces could be replaced by a single force that is applied to this point and is equal to the force of gravity.
To study the movements of an athlete's body, the term general center of gravity (GCG) is often introduced. Basic properties of the center of gravity:
If the body is fixed on an axis passing through the center of gravity, then the force of gravity will not cause it to rotate;
The center of gravity is the point of application of gravity;
In a uniform field, the center of gravity coincides with the center of mass.
Equilibrium is a body position in which it can remain at rest for as long as desired. When a body deviates from its equilibrium position, the forces acting on it change and the balance of forces is disrupted.
There are different types of equilibrium (Fig. 9). It is customary to distinguish three types of equilibrium: stable, unstable and indifferent.
Stable equilibrium (Fig. 9, a) is characterized by the fact that the body returns to its original position when it is deflected. In this case, forces or moments of force arise, tending to return the body to its original position. An example is the position of the body with upper support (for example, hanging on a crossbar), when, with any deviations, the body tends to return to the initial position.
Indifferent equilibrium (Fig. 9, b) is characterized by the fact that when the position of the body changes, no forces or moments of force arise that tend to return the body to its initial position or further remove the body from it. This is a rare occurrence in humans. An example is the state of weightlessness on a spaceship.
Unstable equilibrium (Fig. 9, c) is observed when, with small deviations of the body, forces or moments of force arise that tend to deviate the body even more from the initial position. Such a case can be observed when a person, standing on a support of a very small area (much less than the area of his two legs or even one leg), leans to the side.
Figure 9. Body Balance: stable (a), indifferent (b), unstable (c)
Along with the listed types of equilibrium of bodies, biomechanics considers another type of equilibrium - limited-stable. This type of equilibrium is distinguished by the fact that the body can return to its initial position when deviating from it to a certain limit, for example, determined by the boundary of the support area. If the deviation exceeds this limit, the equilibrium becomes unstable.
The main task in ensuring the balance of the human body is to ensure that the projection of the body's GCM is within the support area. Depending on the type of activity (maintaining a static position, walking, running, etc.) and the requirements for stability, the frequency and speed of corrective influences change, but the processes of maintaining balance are the same.
Distribution of mass in the human body
Body mass and masses of individual segments are very important for various aspects of biomechanics. In many sports, it is necessary to know the distribution of mass in order to develop the correct technique for performing exercises. To analyze the movements of the human body, the segmentation method is used: it is conditionally dissected into certain segments. For each segment, its mass and the position of the center of mass are determined. In table 1 the masses of body parts are determined in relative units.
Table 1. Masses of body parts in relative units
Often, instead of the concept of the center of mass, another concept is used - the center of gravity. In a uniform field of gravity, the center of gravity always coincides with the center of mass. The position of the center of gravity of the link is indicated as its distance from the axis of the proximal joint and is expressed relative to the length of the link, taken as a unit.
In table Figure 2 shows the anatomical position of the centers of gravity of various parts of the body.
Table 2. Centers of gravity of body parts
| Part of the body | Center of gravity position |
| Hip | 0.44 link length |
| Shin | 0.42 link length |
| Shoulder | 0.47 link length |
| Forearm | 0.42 link length |
| Torso | |
| Head | |
| Brush | |
| Foot | |
| Shoulder | 0.47 link length |
| Forearm | 0.42 link length |
| Torso | 0.44 distances from the transverse axis of the shoulder joints to the axis of the hip joints |
| Head | Located in the area of the sella turcica of the sphenoid bone (projection from the front between the eyebrows, from the side - 3.0 - 3.5 above the external auditory canal) |
| Brush | In the region of the head of the third metacarpal bone |
| Foot | On a straight line connecting the calcaneal tubercle of the calcaneus with the end of the second toe at a distance of 0.44 from the first point |
| General center of mass of gravity with a vertical body position | Located at the main stance in the pelvic area, in front of the sacrum |
Let us select an elementary volume dV=dx dy dz in an inhomogeneous solid (Fig. 5.3). The weight of the selected element will be , where is the specific weight at a point of the body with the corresponding coordinates.
The weights of the elements form a system of forces parallel to the applicate axis. Resultant module
element weights is called weight rigid body, and the geometric point of application of the resultant is center of gravity solid body. To calculate these quantities, we use formulas (5.1) and (5.4), replacing summation in them with integration over volume, that is
The quantity in the numerator of formula (5.8) is called the static moment of the weight of a rigid body relative to the coordinate plane.
Obviously, for a homogeneous body, formula (5.8) takes the form
The structure of formulas for calculations is similar.
In this case, the center of gravity of the solid body coincides with the center of its volume.
If one of the dimensions of a solid body is significantly smaller than the other two, the body is called heavy surface. With a constant weight per unit surface area, it is homogeneous. Formulas for calculating the weight and coordinates of the center of gravity are obtained from (5.7) – (5.9) by replacing integrals over the volume with integrals over the surface. In some cases the surface may be flat.
If two dimensions of a solid body are significantly smaller than the third, the body is called heavy line. With a constant weight per unit length of the line, it is homogeneous. Formulas for calculating the weight and coordinates of the center of gravity are obtained from (5.7) – (5.9) by replacing the volume integrals with curvilinear integrals. In some cases the line may be straight.
If a homogeneous solid body has a plane of symmetry, then the center of gravity of the body lies in this plane (the sum of the static moments of elementary weight forces relative to the plane of symmetry is zero).
If a homogeneous solid body has two planes of symmetry, then the center of gravity of the body belongs to the line of intersection of these planes.
If a homogeneous solid body has three planes of symmetry, then the center of gravity of the body is located at the point of their intersection.
If a rigid body can be mentally divided into elements whose weights and positions of centers of gravity are known, then the weight of the rigid body and the position of its center of gravity can be calculated using formulas (5.1) and (5.4). For example, the weight and coordinates of the center of gravity of a ship under construction are calculated.
If the body has cutouts, then they can be counted as negative weight elements.
Note that in the engineering reference literature there is a fairly large number of homogeneous elements (volumetric, flat and curved), for which the weights and positions of the centers of gravity are calculated. The table below shows some of them.
| Element type | Volume (area) of the element | Abscissa c.t. | Ordinate c.t. | Applicata c.t. |
In some situations, the position of the center of gravity of a rigid body can be found from the results of the experiment. For example, when hanging a body on a thread, its center of gravity is located on the line of the thread. By hanging the body by another point that does not lie on the first line, we find the position of the center of gravity of the body as the point of intersection of two lines. Another method used to find the center of gravity of extended bodies is the so-called placing it on “knives” with parallel blades. When the “knives” come together, the center of gravity of the body tends to remain between them and, in the limit, ends up on the line of coincidence of the blades.
In engineering practice, methods that are a combination of calculation and experiment can be used to determine the position of the center of gravity of a body. As an example, let us give the calculation of the distance of the center of gravity of the aircraft, shown in Fig. 5.4., from its front wheel.
In the figure: D is a dynamometer showing the magnitude of the normal pressure force of the front wheel, P is the weight of the aircraft, is the distance from the front wheel to the axis of the rear wheels.
Obviously, the distance of interest from the front wheel to the plane's weight force line can be obtained from the equation of the sum of the moments of forces and P about the axis of the rear wheels, as
Note: if the weight P of the aircraft is not known, then by moving the dynamometer D under the rear wheels, you can obtain the value of the normal pressure force. Then
Example 5.1. For a homogeneous plate that has the shape of a circular sector with an angle of 2 at the apex (see Fig. 5.5), find the position of the center of gravity of the plate.
Let's draw the x-axis so that it is the bisector of angle 2. Then, due to symmetry, the ordinate of the center of gravity is equal to zero, i.e. .
Using two radii, the elementary angle between which is , we select an element on the plate whose area is approximately equal to the area of an isosceles triangle
The abscissa of the center of gravity of the selected triangular element is equal to .
Now we can construct an expression for calculating the abscissa of the center of gravity of a circular sector as
Note: during the calculation it was taken into account that the center of gravity of a homogeneous flat body has the same coordinates on the plane as those of the corresponding flat figure.
Example 5.2. For a thin homogeneous plate of complex shape, the dimensions of which are indicated in Fig. 5.6, find the position of the center of gravity.
Let us mentally divide the plate into three elements: a rectangle, a triangle and a circle. For each of the elements, we find the area and coordinates of the center of gravity:
Then for the plate the coordinates of the center of gravity can be calculated using the formulas:
When calculating, the hole was treated as the attachment of a circle of negative weight.
Center of gravity a geometric point, invariably associated with a solid body, through which the resultant of all gravitational forces acting on the particles of this body passes at any position of the latter in space; it may not coincide with any of the points of a given body (for example, near a ring). If a free body is suspended on threads attached sequentially to different points of the body, then the directions of these threads will intersect at the center of the body. The position of the center of mass of a solid body in a uniform field of gravity coincides with the position of its center of mass (see Center of mass). Breaking the body apart with weights pk, for which the coordinates x k , y k , z k Their center points are known, you can find the coordinates of the center point of the whole body using the formulas:
Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .
Synonyms:See what “Center of Gravity” is in other dictionaries:
The center of mass (center of inertia, barycenter) in mechanics is a geometric point that characterizes the movement of a body or a system of particles as a whole. Contents 1 Definition 2 Centers of mass of homogeneous figures 3 In mechanics ... Wikipedia
A point invariably associated with a solid body through which the resultant of the gravitational forces acting on the particles of this body passes at any position of the body in space. For a homogeneous body that has a center of symmetry (circle, ball, cube, etc.),... ... encyclopedic Dictionary
Geom. a point invariably associated with a solid body through which the resultant force of all gravitational forces acting on the particles of the body passes through it at any position in space; it may not coincide with any of the points of a given body (for example, at ... ... Physical encyclopedia
A point invariably associated with a solid body through which the resultant of the gravitational forces acting on the particles of this body passes at any position of the body in space. For a homogeneous body that has a center of symmetry (circle, ball, cube, etc.),... ... Big Encyclopedic Dictionary
Center of gravity- CENTER OF GRAVITY, the point through which the resultant of the forces of gravity acting on the particles of a solid body passes at any position of the body in space. For a homogeneous body that has a center of symmetry (circle, ball, cube, etc.), the center of gravity is ... Illustrated Encyclopedic Dictionary
CENTER OF GRAVITY, the point at which the weight of a body is concentrated and around which its weight is distributed and balanced. A freely falling object rotates around its center of gravity, which in turn rotates along a trajectory that would be described by a point... ... Scientific and technical encyclopedic dictionary
center of gravity- solid body; center of gravity The center of parallel gravitational forces acting on all particles of a body... Polytechnic terminological explanatory dictionary
Centroid Dictionary of Russian synonyms. center of gravity noun, number of synonyms: 12 main (31) spirit ... Synonym dictionary
CENTER OF GRAVITY- The human body does not have a permanent anatomy. location inside the body, and moves depending on changes in posture; its excursions relative to the spine can reach 20-25 cm. Experimental determination of the position of the central nervous system of the whole body with... ... Great Medical Encyclopedia
The point of application of the resultant forces of gravity (weights) of all individual parts (parts) that make up a given body. If the body is symmetrical with respect to a plane, a straight line or a point, then in the first case the center of gravity lies in the plane of symmetry, in the second on ... ... Technical railway dictionary
center of gravity- The geometric point of a solid body through which the resultant of all gravity forces acting on the particles of this body passes at any position in space [Terminological dictionary of construction in 12 languages (VNIIIS Gosstroy... ... Technical Translator's Guide
Books
- Center of Gravity, A.V. Polyarinov. Alexey Polyarinov’s novel resembles a complex system of lakes. It contains cyberpunk, and the majestic designs of David Mitchell, and Borges, and David Foster Wallace... But its heroes are young journalists,...
Archimedes' first discovery in mechanics was the introduction of the concept of the center of gravity, i.e. proof that in any body there is a single point at which its weight can be concentrated without disturbing the equilibrium state.
The center of gravity of a body is the point of a solid body through which the resultant of all gravitational forces acting on the elementary masses of this body passes at any position in space.
Center of gravity of the mechanical system is the point relative to which the total moment of gravity acting on all bodies of the system is equal to zero.
Simply put, center of gravity- this is the point to which the force of gravity is applied, regardless of the position of the body itself. If the body is homogeneous, center of gravity usually located at the geometric center of the body. Thus, the center of gravity in a homogeneous cube or a homogeneous ball coincides with the geometric center of these bodies.
If the dimensions of the body are small compared to the radius of the Earth, then we can assume that the gravitational forces of all particles of the body form a system of parallel forces. Their resultant is called gravity, and the center of these parallel forces is center of gravity of the body.The coordinates of the body’s center of gravity can be determined using the formulas (Fig. 7.1):
, 
, 
,
Where 
- body weight x i, y i, z i– coordinates of an elementary particle, weight P i;.
Formulas for determining the coordinates of the center of gravity of a body are accurate, strictly speaking, only when the body is divided into an infinite number of infinitely small elementary particles weighing P i. If the number of particles into which the body is mentally divided is finite, then in the general case these formulas will be approximate, since the coordinates x i, y i, z i in this case, they can only be determined with an accuracy of particle sizes. The smaller these particles are, the smaller the error we will make when calculating the coordinates of the center of gravity. Exact expressions can only be reached as a result of passing to the limit, when the size of each particle tends to zero, and their number increases indefinitely. As is known, such a limit is called a definite integral. Therefore, the actual determination of the coordinates of the centers of gravity of bodies in the general case requires the replacement of sums with their corresponding integrals and the use of methods of integral calculus.
If the mass inside a solid body or mechanical system is distributed non-uniformly, then the center of gravity shifts to the part where it is heavier.
The center of gravity of a body may not always even be located inside the body itself. So, for example, the center of gravity of a boomerang is somewhere in the middle between the ends of the boomerang, but outside the body of the boomerang itself.
For securing loads, the position of the center of gravity is very important. It is at this point that the forces of gravity and inertial forces acting on the load during movement are applied. The higher the center of gravity of a body or mechanical system, the more prone it is to tipping over.
The center of gravity of the body coincides with the center of mass.
