Depending on the shape of the trajectory, movement can be divided into rectilinear and curvilinear. Most often you encounter curvilinear movements when the trajectory is represented as a curve. An example of this type of motion is the path of a body thrown at an angle to the horizon, the movement of the Earth around the Sun, planets, and so on.

Picture 1 . Trajectory and movement in curved motion

Curvilinear movement called a movement whose trajectory is a curved line. If a body moves along a curved path, then the displacement vector s → is directed along the chord, as shown in Figure 1, and l is the length of the path. The direction of the instantaneous speed of movement of the body goes tangentially at the same point of the trajectory where at this moment the moving object is located, as shown in Figure 2.

Figure 2. Instantaneous speed during curved motion

Definition 2

Curvilinear movement material point called uniform when the velocity module is constant (circular motion), and uniformly accelerated when the direction and velocity module are changing (movement of an thrown body).

Curvilinear motion is always accelerated. This is explained by the fact that even with an unchanged velocity module and a changed direction, acceleration is always present.

In order to study the curvilinear motion of a material point, two methods are used.

The path is divided into separate sections, at each of which it can be considered straight, as shown in Figure 3.

Figure 3. Partitioning curvilinear motion into translational ones

Now the law of rectilinear motion can be applied to each section. This principle is allowed.

The most convenient solution method is considered to represent the path as a set of several movements along circular arcs, as shown in Figure 4. The number of partitions will be much less than in the previous method, in addition, the movement along the circle is already curvilinear.

Figure 4. Partitioning curvilinear motion into motion along circular arcs

Note 1

To record curvilinear motion, you must be able to describe motion in a circle, voluntary movement represented as sets of movements along the arcs of these circles.

The study of curvilinear motion includes the compilation of a kinematic equation that describes this motion and allows one to determine all the characteristics of the motion based on the available initial conditions.

Example 1

Given a material point moving along a curve, as shown in Figure 4. The centers of circles O 1, O 2, O 3 are located on the same straight line. Need to find displacement
s → and path length l while moving from point A to B.

Solution

By condition, we have that the centers of the circle belong to the same straight line, hence:

s → = R 1 + 2 R 2 + R 3 .

Since the trajectory of movement is the sum of semicircles, then:

l ~ A B = π R 1 + R 2 + R 3 .

Answer: s → = R 1 + 2 R 2 + R 3, l ~ A B = π R 1 + R 2 + R 3.

Example 2

The dependence of the distance traveled by the body on time is given, represented by the equation s (t) = A + B t + C t 2 + D t 3 (C = 0.1 m / s 2, D = 0.003 m / s 3). Calculate after what period of time after the start of movement the acceleration of the body will be equal to 2 m / s 2

Solution

Answer: t = 60 s.

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Uniformly accelerated curvilinear motion

Curvilinear movements are movements whose trajectories are not straight, but curved lines. Planets and river waters move along curvilinear trajectories.

Curvilinear motion is always motion with acceleration, even if the absolute value of the velocity is constant. Curvilinear movement with constant acceleration always occurs in the plane in which the acceleration vectors and initial velocities of the point are located. In the case of curvilinear motion with constant acceleration in the xOy plane, the projections vx and vy of its velocity on the Ox and Oy axes and the x and y coordinates of the point at any time t are determined by the formulas

Uneven movement. Rough speed

No body moves all the time constant speed. When the car starts moving, it moves faster and faster. It can move steadily for a while, but then it slows down and stops. In this case, the car travels different distances in the same time.

Movement in which a body travels unequal lengths of path in equal intervals of time is called uneven. With such movement, the speed does not remain unchanged. In this case, we can only talk about average speed.

Average speed shows the distance a body travels per unit time. It is equal to the ratio of the displacement of the body to the time of movement. Average speed, like the speed of a body during uniform motion, is measured in meters divided by a second. In order to characterize motion more accurately, instantaneous speed is used in physics.

The speed of a body at a given moment in time or at a given point in the trajectory is called instantaneous speed. Instantaneous speed is a vector quantity and is directed in the same way as the displacement vector. You can measure instantaneous speed using a speedometer. In the International System, instantaneous speed is measured in meters divided by second.

point movement speed uneven

Movement of a body in a circle

Curvilinear motion is very common in nature and technology. It is more complex than a straight line, since there are many curved trajectories; this movement is always accelerated, even when the velocity module does not change.

But movement along any curved path can be approximately represented as movement along the arcs of a circle.

When a body moves in a circle, the direction of the velocity vector changes from point to point. Therefore, when they talk about the speed of such movement, they mean instantaneous speed. The velocity vector is directed tangentially to the circle, and the displacement vector is directed along the chords.

Uniform circular motion is a motion during which the modulus of the motion velocity does not change, only its direction changes. The acceleration of such motion is always directed towards the center of the circle and is called centripetal. In order to find the acceleration of a body moving in a circle, it is necessary to divide the square of the speed by the radius of the circle.

In addition to acceleration, the motion of a body in a circle is characterized by the following quantities:

The period of rotation of a body is the time during which the body makes one complete revolution. The rotation period is designated by the letter T and is measured in seconds.

The frequency of rotation of a body is the number of revolutions per unit time. Is the rotation speed indicated by a letter? and is measured in hertz. In order to find the frequency, you need to divide one by the period.

Linear speed is the ratio of the movement of a body to time. In order to find the linear speed of a body in a circle, it is necessary to divide the circumference by the period (the circumference is equal to 2? multiplied by the radius).

Angular velocity - physical quantity, equal to the ratio of the angle of rotation of the radius of the circle along which the body moves to the time of movement. Angular velocity is indicated by a letter? and is measured in radians divided per second. Can you find the angular velocity by dividing 2? for a period of. Angular velocity and linear velocity among themselves. In order to find the linear speed, it is necessary to multiply the angular speed by the radius of the circle.

Figure 6. Circular motion, formulas.

We know that when straight motion the direction of the velocity vector always coincides with the direction of movement. What can be said about the direction of velocity and displacement during curved motion? To answer this question, we will use the same technique that we used in the previous chapter when studying the instantaneous speed of rectilinear motion.

Figure 56 shows a certain curved trajectory. Let us assume that a body moves along it from point A to point B.

In this case, the path traveled by the body is an arc A B, and its displacement is a vector. Of course, one cannot assume that the speed of the body during movement is directed along the displacement vector. Let us draw a series of chords between points A and B (Fig. 57) and imagine that the body’s movement occurs precisely along these chords. On each of them the body moves rectilinearly and the velocity vector is directed along the chord.

Let us now make our straight sections (chords) shorter (Fig. 58). As before, on each of them the velocity vector is directed along the chord. But it is clear that the broken line in Figure 58 is already more similar to a smooth curve.

It is clear, therefore, that by continuing to reduce the length of the straight sections, we will, as it were, pull them into points and the broken line will turn into a smooth curve. The speed at each point of this curve will be directed tangentially to the curve at this point (Fig. 59).

The speed of movement of a body at any point on a curvilinear trajectory is directed tangentially to the trajectory at that point.

The fact that the speed of a point during curvilinear movement is really directed along a tangent is convinced by, for example, observation of the operation of the gochnla (Fig. 60). If you press the ends of a steel rod against a rotating grindstone, the hot particles coming off the stone will be visible in the form of sparks. These particles fly at the speed at which

they possessed at the moment of separation from the stone. It is clearly seen that the direction of the sparks always coincides with the tangent to the circle at the point where the rod touches the stone. The splashes from the wheels of a skidding car also move tangentially to the circle (Fig. 61).

Thus, the instantaneous velocity of a body at different points of a curvilinear trajectory has different directions, as shown in Figure 62. The magnitude of the velocity can be the same at all points of the trajectory (see Figure 62) or vary from point to point, from one moment in time to another (Fig. 63).