§4 Acceleration.
Tangential and normal components of the acceleration

Acceleration - is a vector quantity that characterizes the rate of change of velocity of the moving body in magnitude and direction.

Point average acceleration in the time interval Δt is the vector aav increment equal to the ratio of the velocity vector Δv to the time interval Δt

Acceleration (instantaneous acceleration) point is called a vector quantity, which is equal to the first derivative of the velocity v in time (or the second derivative of the radius vector of the time t)

Acceleration of a point at time t is equal to the limit of the average acceleration of  when

In a Cartesian coordinate system vector  can be written in terms of its coordinates


Module of the acceleration vector

Vector  can be expressed as the sum of two components:
- tangential component of acceleration is tangential to the trajectory of the point and is equal to

where the vector - the unit vector of the tangent drawn at the point of the trajectory and direction of the velocity



Vectors and  collinear with uniformly accelerated motion;  at  ie at uniformly retarded motion.

Tangential acceleration - characterizes the quickness of change of velocity vector modulus of (measures change in velocity magnitude).

For uniform motion


- normal component of acceleration (normal acceleration) along the normal to the trajectory and the given point in the direction of the center of curvature of the trajectory. Curved trajectory can be represented as a set of elementary sections, each of which can be seen as a circular arc of radius R (called the radius of curvature of a circle of a given point of the trajectory)

Normal acceleration characterizes the speed change of direction of the velocity vector (characterizes the change in the direction of the velocity).
Full acceleration module:

The classification depends on the movements of the tangential and normal components:

  1.  - constant motion;
  2.  - uniformly accelerated motion;
  3.  - uniformly retarded motion;
  4.  - linear motion with variable acceleration;
  5.  - uniform circular motion;
  6.  -  uniform curvilinear motion;
  7.  - curved uniformly accelerated motion;
  8.  - curvilinea uniformly retarded motion;
  9.   - curvilinear motion with variable acceleration.



§ 5 The kinematics of rotational motion

Rotation of the body at a certain angle
φ can be described by a vector of length φ, and the direction coincides with the axis of rotation is determined by the rule of the right screw (corkscrew, right hand):

Four fingers of the right hand – on the direction of rotation, bent thumb indicates the direction of the vector .

The direction of the rotation vector φ, associated with the direction of rotation right-hand rule. Such vectors are referred to as the axial or pseudo-to distinguish them from ordinary (sometimes called field) vectors. Called the angular velocity vector  which is numerically equal to the first derivative of the angle of rotation on time t and is directed along a fixed axis on the right hand rule.

Angular velocity, as , is an axial vector. Axial vectors do not have certain points of application, they can be deposited from any point on the axis of rotation. Often they are put off by a stationary point of the rotation axis, taken simultaneously as the origin of the reference framet. Rotation of the body is called uniform if .

 divide by Δt

Speed points as opposed to the angular velocity  of a body, called the linear speed. It is perpendicular to both the axis of rotation (i.e., the vector), and the radius - vector R, drawing to a point P from the center of the circle and about equal to the vector product:


Uniform rotation can be a characterization of the rotation period T, which are defined as the time in which the body makes one revolution, i.e. rotated by an angle. Then

- relationship of the angular velocity with the circulation period.

rotating speed - number of revolutions per unit of time.

In the case of variable rotational motion angular velocity  of material point changes in both magnitude and direction. To characterize the rate of change of the angular velocity in irregular rotation around a fixed axis vector is introduced - angular acceleration of the body is equal to the first derivative of its angular velocity on time

Vector  is also an axial (or pseudovector). Vectors and same direction for accelerated rotation


and  opposite directions during decelerated rotation

Acceleration  arbitrary point P of the body in contrast to the angular acceleration of the body is called linear acceleration.

For uniformly accelerated rotational motion can be written:


Relationship between linear and angular values:




The time dependence of

Path (displacement)




rotation vector





linear speed

angular velocity


linear acceleration

Angular accele-





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