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1.7. Curved motion. Tangential and normal acceleration

Lecture



Curvilinear motion is a motion whose trajectory is a curved line (for example, a circle, ellipse, hyperbola, parabola). An example of curvilinear motion is the movement of the planets, the end of the clock hand on the dial, etc. In the general case, the speed during curvilinear motion varies in magnitude and direction.

Curvilinear motion of a material point is considered uniform motion if the velocity modulus is constant (for example, uniform motion along a circle), and uniformly accelerated if the modulus and direction of velocity changes (for example, the motion of a body thrown at an angle to the horizon).

  1.7.  Curved motion.  Tangential and normal acceleration

Fig. 1.19. Trajectory and displacement vector during curvilinear motion.

When moving along a curved path, the displacement vector   1.7.  Curved motion.  Tangential and normal acceleration sent along the chord (Fig. 1.19), and l - the length of the trajectory. The instantaneous velocity of the body (that is, the velocity of the body at a given point of the trajectory) is tangential at the point of the trajectory where the moving body is at a given moment (Fig. 1.20).

  1.7.  Curved motion.  Tangential and normal acceleration

Fig. 1.20. Instant speed with curvilinear motion.

Curvilinear motion is always accelerated motion. That is, acceleration during curvilinear motion is always present, even if the velocity modulus does not change, only the direction of velocity changes. The change in the magnitude of velocity per unit of time is tangential acceleration:

  1.7.  Curved motion.  Tangential and normal acceleration

or

  1.7.  Curved motion.  Tangential and normal acceleration

Where v τ , v 0 - values ​​of speeds at the moment of time t 0 + Δt and t 0, respectively.

Tangential acceleration at a given point of the trajectory in the direction coincides with the direction of the velocity of the body or opposite to it.

Normal acceleration is the change in speed in a direction per unit of time:

  1.7.  Curved motion.  Tangential and normal acceleration

Normal acceleration is directed along the radius of curvature of the trajectory (to the axis of rotation). Normal acceleration perpendicular to the direction of speed.

Centripetal acceleration is normal acceleration with uniform motion in a circle.

Full acceleration with a uniformly curved body motion is equal to:

  1.7.  Curved motion.  Tangential and normal acceleration

The movement of the body along a curvilinear trajectory can be approximated as a movement along the arcs of certain circles (Fig. 1.21).

  1.7.  Curved motion.  Tangential and normal acceleration

Fig. 1.21. Body movement with curvilinear motion.

In the case of rectilinear motion, the vectors of speed and acceleration coincide with the direction of the trajectory. Consider the motion of a material point along a curved plane trajectory. The velocity vector at any point of the trajectory is tangential to it. Assume that in the TM trajectory the speed was   1.7.  Curved motion.  Tangential and normal acceleration , and in M. 1 became   1.7.  Curved motion.  Tangential and normal acceleration . At the same time, we consider that the time span when a point passes on the path   1.7.  Curved motion.  Tangential and normal acceleration from M to M 1 is so small that the change in acceleration in magnitude and direction can be neglected. To find the velocity change vector   1.7.  Curved motion.  Tangential and normal acceleration It is necessary to determine the vector difference:

  1.7.  Curved motion.  Tangential and normal acceleration

For this we will transfer   1.7.  Curved motion.  Tangential and normal acceleration parallel to itself, combining its beginning with point M. The difference of two vectors is equal to the vector connecting their ends   1.7.  Curved motion.  Tangential and normal acceleration equal to the side of the speaker   1.7.  Curved motion.  Tangential and normal acceleration MAC, built on the velocity vectors, as on the sides. Expand the vector   1.7.  Curved motion.  Tangential and normal acceleration into two components AB and AD, and both respectively   1.7.  Curved motion.  Tangential and normal acceleration and   1.7.  Curved motion.  Tangential and normal acceleration . Thus, the velocity change vector   1.7.  Curved motion.  Tangential and normal acceleration equal to the vector sum of two vectors:

  1.7.  Curved motion.  Tangential and normal acceleration

By definition:

  1.7.  Curved motion.  Tangential and normal acceleration (1.15)

Tangential acceleration   1.7.  Curved motion.  Tangential and normal acceleration characterizes the rapidity of change in the speed of movement by the numerical value and is directed tangentially to the trajectory.

Consequently

  1.7.  Curved motion.  Tangential and normal acceleration (1.16)

Normal acceleration   1.7.  Curved motion.  Tangential and normal acceleration characterizes the rate of change of speed in the direction. Calculate the vector:

  1.7.  Curved motion.  Tangential and normal acceleration

For this we draw a perpendicular through the points M and M1 to the tangents to the trajectory (fig. 1.4) the intersection point we denote by O. For a sufficiently small   1.7.  Curved motion.  Tangential and normal acceleration The curved path section can be considered a part of a circle of radius R. The triangles MOM1 and MBC are similar, because they are isosceles triangles with the same angles at the vertices. Therefore:

  1.7.  Curved motion.  Tangential and normal acceleration

or

  1.7.  Curved motion.  Tangential and normal acceleration

But   1.7.  Curved motion.  Tangential and normal acceleration , then:

  1.7.  Curved motion.  Tangential and normal acceleration

Going to the limit at   1.7.  Curved motion.  Tangential and normal acceleration and given that   1.7.  Curved motion.  Tangential and normal acceleration , we find:

  1.7.  Curved motion.  Tangential and normal acceleration ,

  1.7.  Curved motion.  Tangential and normal acceleration (1.17)

Since when   1.7.  Curved motion.  Tangential and normal acceleration angle   1.7.  Curved motion.  Tangential and normal acceleration , the direction of this acceleration coincides with the direction of the normal to speed   1.7.  Curved motion.  Tangential and normal acceleration i.e. acceleration vector   1.7.  Curved motion.  Tangential and normal acceleration perpendicular   1.7.  Curved motion.  Tangential and normal acceleration . Therefore, this acceleration is often called centripetal.

Full acceleration is determined by the vector sum of the tangential normal accelerations (1.15). Since the vectors of these accelerations are mutually perpendicular, the total acceleration modulus is equal to:

  1.7.  Curved motion.  Tangential and normal acceleration (1.18)

The direction of the full acceleration is determined by the angle between the vectors   1.7.  Curved motion.  Tangential and normal acceleration and   1.7.  Curved motion.  Tangential and normal acceleration :

  1.7.  Curved motion.  Tangential and normal acceleration


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Physical foundations of mechanics

Terms: Physical foundations of mechanics