The movement of a material point along a curved path is always accelerated, since even if the speed does not change in numerical value, it always changes in direction.
In general, acceleration during curvilinear motion can be represented as a vector sum of tangential (or tangential) acceleration t and normal acceleration n: =t+n- rice. 1.4.
Tangential acceleration characterizes the rate of change in velocity modulo. The value of this acceleration will be:
Normal acceleration characterizes the rate of change in speed in direction. The numerical value of this acceleration, where r- radius of the contacting circle, i.e. a circle drawn through three infinitely close points B¢ , A, B, lying on the curve (Fig. 1.5). Vector n directed along the normal to the trajectory to the center of curvature (the center of the osculating circle).
Numerical value of total acceleration
where is the angular velocity.
where is the angular acceleration.
Angular acceleration is numerically equal to the change in angular velocity per unit time.
In conclusion, we present a table that establishes an analogy between the linear and angular kinematic parameters of motion.
End of work -
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Basic SI units
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Mechanics
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Newton's laws
Dynamics is a branch of mechanics that studies the movement of material bodies under the influence of forces applied to them. Mechanics is based on Newton's laws. Newton's first law
Law of conservation of momentum
Let's consider the derivation of the law of conservation of momentum based on Newton's second and third laws.
Relationship between work and change in kinetic energy
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Relationship between work and change in potential energy
Rice. 3.4 We will establish this connection using the example of the work of gravity
Law of conservation of mechanical energy
Let us consider a closed conservative system of bodies. This means that the bodies of the system are not affected by external forces, and internal forces are conservative in nature. Full mechanical
Collisions
Let us consider an important case of interaction of solid bodies - collisions. Collision (impact) is the phenomenon of a finite change in the velocities of solid bodies over very short periods of time when they are not
Basic law of the dynamics of rotational motion
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Law of conservation of angular momentum
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Gyroscope
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General characteristics of oscillatory processes. Harmonic vibrations
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Oscillations of a spring pendulum
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Energy of harmonic vibration
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Addition of harmonic vibrations of the same direction
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Damped oscillations
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Forced vibrations
In real conditions, an oscillating system gradually loses energy to overcome frictional forces, so the oscillations are damped. For the oscillations to be undamped, it is necessary somehow
Elastic (mechanical) waves
The process of propagation of disturbances in a substance or field, accompanied by the transfer of energy, is called a wave. Elastic waves - the process of mechanical propagation in an elastic medium
Wave interference
Interference is the phenomenon of superposition of waves from two coherent sources, as a result of which a redistribution of wave intensity occurs in space, i.e. interference occurs
Standing waves
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Doppler effect in acoustics
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Barometric formula
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Boltzmann distribution
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The first law of thermodynamics and its application to isoprocesses
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Number of degrees of freedom. Internal energy of an ideal gas
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Adiabatic process
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Reversible and irreversible processes. Circular processes (cycles). Operating principle of a heat engine
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Ideal Carnot heat engine
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Second law of thermodynamics
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Electrostatic field energy
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Main characteristics of current
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Ohm's law for a homogeneous section of a chain
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Joule-Lenz law
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Kinematics of a point, kinematics of a rigid body, translational motion, rotational motion, plane-parallel motion, theorem on velocity projections, instantaneous center of velocities, determination of the velocity and acceleration of points of a plane body, complex motion of a point
ContentRigid body kinematics
To uniquely determine the position of a rigid body, you need to specify three coordinates (x A , y A , z A ) one of the points A of the body and three rotation angles. Thus, the position of a rigid body is determined by six coordinates. That is, a rigid body has six degrees of freedom.
In the general case, the dependence of the coordinates of points on a rigid body relative to a fixed coordinate system is determined by rather cumbersome formulas. However, the velocities and accelerations of points are determined quite simply. To do this, you need to know the dependence of the coordinates on time of one, arbitrarily selected, point A and the angular velocity vector. Differentiating with respect to time, we find the speed and acceleration of point A and the angular acceleration of the body:
; ; .
Then the speed and acceleration of a point of a body with a radius vector are determined by the formulas:
(1)
;
(2)
.
Here and below, products of vectors in square brackets mean vector products.
Note that the angular velocity vector is the same for all points of the body. It does not depend on the coordinates of body points. Also the angular acceleration vector is the same for all points of the body.
See formula output (1) And (2) on page: Velocity and acceleration of points of a rigid body > > >
Translational motion of a rigid body
During translational motion, the angular velocity is zero. The velocities of all points of the body are equal. Any straight line drawn in the body moves, remaining parallel to its initial direction. Thus, to study the motion of a rigid body during translational motion, it is enough to study the motion of any one point of this body. See section.
Uniformly accelerated motion
Let us consider the case of uniformly accelerated motion. Let the projection of the acceleration of a point of the body onto the x axis be constant and equal to a x. Then the projection of velocity v x and x - the coordinate of this point depend on time t according to the law:
v x = v x 0 + a x t;
,
where v x 0
and x 0
- speed and coordinate of the point at the initial moment of time t = 0
.
Rotational motion of a rigid body
Consider a body that rotates around a fixed axis. Let us choose a fixed coordinate system Oxyz with center at point O. Let's direct the z axis along the rotation axis. We assume that the z-coordinates of all points of the body remain constant. Then the movement occurs in the xy plane. Angular velocity ω and angular acceleration ε are directed along the z axis:
; .
Let φ be the angle of rotation of the body, which depends on time t. Differentiating with respect to time, we find projections of angular velocity and angular acceleration to the z axis:
;
.
Let's consider the movement of a point M, which is located at a distance r from the axis of rotation. The trajectory of motion is a circle (or arc of a circle) of radius r.
Point speed:
v = ωr.
The velocity vector is directed tangentially to the trajectory.
Tangential acceleration:
a τ = ε r .
The tangential acceleration is also directed tangentially to the trajectory.
Normal acceleration:
.
It is directed towards the axis of rotation O.
Full acceleration:
.
Since the vectors and are perpendicular to each other, then acceleration module:
.
Uniformly accelerated motion
In the case of uniformly accelerated motion, in which the angular acceleration is constant and equal to ε, the angular velocity ω and the angle of rotation φ change with time t according to the law:
ω = ω 0 + ε t;
,
where ω 0
and φ 0
- angular velocity and angle of rotation at the initial moment of time t = 0
.
Plane-parallel motion of a rigid body
Plane-parallel or flat is the movement of a rigid body in which all its points move parallel to some fixed plane. Let's choose a rectangular coordinate system Oxyz. We will place the x and y axes in the plane in which the points of the body move. Then all z - coordinates of the points of the body remain constant, z - components of velocities and accelerations are equal to zero. The vectors of angular velocity and angular acceleration, on the contrary, are directed along the z axis. Their x and y components are zero.
The projections of the velocities of two points of a rigid body onto an axis passing through these points are equal to each other.
v A cos α = v B cos β.
Instantaneous velocity center
Instantaneous velocity center is the point of a plane figure whose velocity is currently zero.
To determine the position of the instantaneous center of velocities P of a flat figure, you only need to know the directions of the velocities and its two points A and B. To do this, draw a straight line through point A perpendicular to the direction of velocity. Through point B we draw a straight line perpendicular to the direction of velocity. The point of intersection of these lines is the instantaneous center of velocities P. Angular velocity of body rotation:
.
If the velocities of two points are parallel to each other, then ω = 0 . The velocities of all points of the body are equal to each other (at a given moment in time).
If the speed of any point A of a flat body and its angular speed ω are known, then the speed of an arbitrary point M is determined by the formula (1)
, which can be represented as the sum of translational and rotational motion:
,
where is the speed of rotational motion of point M relative to point A. That is, the speed that point M would have when rotating in a circle of radius |AM| with angular velocity ω if point A were stationary.
Relative velocity module:
v MA = ω |AM| .
The vector is directed tangent to the circle of radius |AM| with center at point A.
Determination of accelerations of points of a flat body is carried out using the formula (2)
. The acceleration of any point M is equal to the vector sum of the acceleration of some point A and the acceleration of point M during rotation around point A, considering point A stationary:
.
can be decomposed into tangential and normal accelerations:
.
Tangential acceleration is directed tangentially to the trajectory. Normal acceleration is directed from point M to point A. Here ω and ε are the angular velocity and angular acceleration of the body.
Complex point movement
Let O 1 x 1 y 1 z 1- fixed rectangular coordinate system. The speed and acceleration of point M in this coordinate system will be called absolute speed and absolute acceleration.
Let Oxyz be a moving rectangular coordinate system, say, rigidly connected to a certain rigid body moving relative to the system O 1 x 1 y 1 z 1. The speed and acceleration of point M in the Oxyz coordinate system will be called relative speed and relative acceleration. Let be the angular velocity of rotation of the system Oxyz relative to O 1 x 1 y 1 z 1.
Let's consider a point that, at a given moment in time, coincides with point M and is motionless relative to the Oxyz system (a point rigidly connected to a solid body). Velocity and acceleration of such a point in the coordinate system O 1 x 1 y 1 z 1 we will call it portable speed and portable acceleration.
Velocity addition theorem
The absolute speed of a point is equal to the vector sum of the relative and portable speeds:
.
Acceleration addition theorem (Coriolis theorem)
The absolute acceleration of a point is equal to the vector sum of the relative, transport and Coriolis accelerations:
,
Where
- Coriolis acceleration.
References:
S. M. Targ, Short course in theoretical mechanics, “Higher School”, 2010.
For example, a car that starts moving moves faster as it increases its speed. At the point where the motion begins, the speed of the car is zero. Having started moving, the car accelerates to a certain speed. If you need to brake, the car will not be able to stop instantly, but over time. That is, the speed of the car will tend to zero - the car will begin to move slowly until it stops completely. But physics does not have the term “slowdown”. If a body moves, decreasing speed, this process is also called acceleration, but with a “-” sign.
Medium acceleration is called the ratio of the change in speed to the period of time during which this change occurred. Calculate the average acceleration using the formula:
where is it . The direction of the acceleration vector is the same as the direction of change in speed Δ = - 0
where 0 is the initial speed. At a moment in time t 1(see figure below) at the body 0. At a moment in time t 2 the body has speed. Based on the rule of vector subtraction, we determine the vector of speed change Δ = - 0. From here we calculate the acceleration:
.
In the SI system unit of acceleration called 1 meter per second per second (or meter per second squared):
.
A meter per second squared is the acceleration of a rectilinearly moving point, at which the speed of this point increases by 1 m/s in 1 second. In other words, acceleration determines the degree of change in the speed of a body in 1 s. For example, if the acceleration is 5 m/s2, then the speed of the body increases by 5 m/s every second.
Instantaneous acceleration of a body (material point) at a given moment in time is a physical quantity that is equal to the limit to which the average acceleration tends as the time interval tends to 0. In other words, this is the acceleration developed by the body in a very short period of time:
.
Acceleration has the same direction as the change in speed Δ in extremely short periods of time during which the speed changes. The acceleration vector can be specified using projections onto the corresponding coordinate axes in a given reference system (projections a X, a Y, a Z).
With accelerated linear motion, the speed of the body increases in absolute value, i.e. v 2 > v 1 , and the acceleration vector has the same direction as the velocity vector 2 .
If the speed of a body decreases in absolute value (v 2< v 1), значит, у вектора ускорения направление противоположно направлению вектора скорости 2 . Другими словами, в таком случае наблюдаем slowing down(acceleration is negative, and< 0). На рисунке ниже изображено направление векторов ускорения при прямолинейном движении тела для случая ускорения и замедления.
If movement occurs along a curved path, then the magnitude and direction of the speed changes. This means that the acceleration vector is depicted as two components.
Tangential (tangential) acceleration they call that component of the acceleration vector that is directed tangentially to the trajectory at a given point of the trajectory of movement. Tangential acceleration describes the degree of change in speed modulo during curvilinear motion.
U tangential acceleration vectorτ (see figure above) the direction is the same as that of linear speed or opposite to it. Those. the tangential acceleration vector is in the same axis with the tangent circle, which is the trajectory of the body.
Tangential (tangential) acceleration – this is the component of the acceleration vector directed along the tangent to the trajectory at a given point of the movement trajectory. Tangential acceleration characterizes the change in speed modulo during curvilinear motion.
Figure 1 – Tangential acceleration
The direction of the tangential acceleration vector coincides with the direction of linear velocity or is opposite to it, from Fig. 1. That is, the tangential acceleration vector lies on the same axis with the tangent circle, which is the trajectory of the body.
Normal acceleration is the component of the acceleration vector directed along the normal to the trajectory of motion at a given point on the trajectory of the body. That is, the normal acceleration vector is perpendicular to the linear speed of movement, shown in Fig. 1. Normal acceleration characterizes the change in speed in direction and is denoted by n. The normal acceleration vector is directed along the radius of curvature of the trajectory.
Full acceleration in curvilinear motion, it consists of tangential and normal accelerations according to the rule of vector addition and is determined by the formula:
(9)
(10)
The direction of total acceleration is also determined by the vector addition rule:
(11)
1.1.5 Translational and rotational motion of an absolutely rigid body
The movement of the body is considered progressive, if any straight line segment rigidly connected to the body constantly moves parallel to itself. During translational motion, all points of the body make the same movements, travel the same paths, have equal speeds and accelerations, and describe the same trajectories.
Rotation of a rigid body around a fixed axis- a movement in which all points of the body describe circles, the centers of which are on the same straight line, perpendicular to the planes of these circles. This straight line itself is the axis of rotation.
When a body rotates, the radius of the circle described by a point of this body will rotate through a certain angle over a time interval. Due to the invariance of the relative position of the points of the body, the radii of the circles described by any other points of the body rotate through the same angle during the same time. This angle is a value that characterizes the rotational motion of the entire body as a whole. From this we can conclude that to describe the rotational motion of an absolutely rigid body around a fixed axis, you need to know only one variable - the angle through which the body will rotate in a certain time.
The relationship between linear and angular velocities for each point of a rigid body is given by the formula:
(12)
Tangential acceleration characterizes the change in speed in absolute value (magnitude) and is directed tangentially to the trajectory:
,
Where 
derivative of the velocity module, 
unit tangent vector, coinciding in direction with the speed.
Normal acceleration characterizes the change in speed in direction and is directed along the radius of curvature to the center of curvature of the trajectory at a given point:
,
where R is the radius of curvature of the trajectory, 
unit normal vector.
The magnitude of the acceleration vector can be found using the formula
.
1.3. The main task of kinematics
The main task of kinematics is to find the law of motion of a material point. For this, the following relationships are used:
;
;
;
;
.
Special cases of rectilinear motion:
1) uniform linear motion: ;
2) uniform linear motion: 
.
1.4. Rotational motion and its kinematic characteristics
During rotational motion, all points of the body move in circles, the centers of which lie on the same straight line, called the axis of rotation. To characterize the rotational motion, the following kinematic characteristics are introduced (Fig. 3).
Angular movement
vector numerically equal to the angle of rotation of the body 
during 
and directed along the axis of rotation so that, looking along it, the rotation of the body is observed to occur clockwise.
Angular velocity
characterizes the speed and direction of rotation of the body, is equal to the derivative of the angle of rotation with respect to time and is directed along the axis of rotation as angular displacement.
P 
For rotational motion, the following formulas are valid:
;
;
.
Angular acceleration
characterizes the rate of change in angular velocity over time, equal to the first derivative of the angular velocity and directed along the axis of rotation:
;
;
.
Addiction 
expresses the law of body rotation.
With uniform rotation: = 0, = const, = t.
With uniform rotation: = const, 
,
.
To characterize uniform rotational motion, the rotation period and rotation frequency are used.
Rotation period T is the time of one revolution of a body rotating at a constant angular velocity.
Rotation frequency – the number of revolutions made by the body per unit of time.
Angular velocity can be expressed as follows:
.
Relationship between angular and linear kinematic characteristics (Fig. 4):
2. Dynamics of translational and rotational movements
Newton's laws Newton's first law: every body is in a state of rest or uniform rectilinear motion until the influence of other bodies takes it out of this state.
Bodies that are not subject to external influences are called free bodies. The reference system associated with a free body is called an inertial reference system (IRS). In relation to it, any free body will move uniformly and rectilinearly or be at rest. From the relativity of motion it follows that a reference system moving uniformly and rectilinearly with respect to the ISO is also an ISO. ISOs play an important role in all branches of physics. This is due to Einstein's principle of relativity, according to which the mathematical form of any physical law must have the same form in all inertial frames of reference.
The basic concepts used in the dynamics of translational motion include force, body mass, and momentum of the body (system of bodies).
By force is a vector physical quantity that is a measure of the mechanical action of one body on another. Mechanical action occurs both through direct contact of interacting bodies (friction, support reaction, weight, etc.), and through force field existing in space (gravity, Coulomb forces, etc.). Force 
characterized by module, direction and point of application.
Simultaneous action of several forces on the body 
,
,...,
can be replaced by the action of the resultant (resultant) force 
:
=
+
+...+
=
.
Mass of a body is a scalar quantity that is a measure inertia bodies. Under inertia refers to the property of material bodies to maintain their speed unchanged in the absence of external influences and change it gradually (i.e. with finite acceleration) under the influence of force.
Impulse body (material point) is a vector physical quantity equal to the product of the body’s mass and its speed: 
.
The momentum of a system of material points is equal to the vector sum of the momentum of the points that make up the system: 
.
Newton's second law: the rate of change of momentum of a body is equal to the force acting on it:
.
If the mass of the body remains constant, then the acceleration acquired by the body relative to the inertial frame of reference is directly proportional to the force acting on it and inversely proportional to the mass of the body:
.
