Circular movement. Movement of a body in a circle with a constant speed in absolute value Finding the speed of a body when moving in a circle

In this lesson, we will consider curvilinear motion, namely the uniform motion of a body in a circle. We will learn what linear speed is, centripetal acceleration when a body moves in a circle. We also introduce quantities that characterize the rotational motion (rotation period, rotation frequency, angular velocity), and connect these quantities with each other.

By uniform motion in a circle is understood that the body rotates through the same angle for any identical period of time (see Fig. 6).

Rice. 6. Uniform circular motion

That is, the module of instantaneous speed does not change:

This speed is called linear.

Although the modulus of the speed does not change, the direction of the speed changes continuously. Consider the velocity vectors at the points A and B(see Fig. 7). They are directed in different directions, so they are not equal. If subtracted from the speed at the point B point speed A, we get a vector .

Rice. 7. Velocity vectors

The ratio of the change in speed () to the time during which this change occurred () is acceleration.

Therefore, any curvilinear motion is accelerated.

If we consider the velocity triangle obtained in Figure 7, then with a very close arrangement of points A and B to each other, the angle (α) between the velocity vectors will be close to zero:

It is also known that this triangle is isosceles, so the modules of velocities are equal (uniform motion):

Therefore, both angles at the base of this triangle are indefinitely close to:

This means that the acceleration that is directed along the vector is actually perpendicular to the tangent. It is known that a line in a circle perpendicular to a tangent is a radius, so acceleration is directed along the radius towards the center of the circle. This acceleration is called centripetal.

Figure 8 shows the triangle of velocities discussed earlier and an isosceles triangle (two sides are the radii of a circle). These triangles are similar, since they have equal angles formed by mutually perpendicular lines (the radius, like the vector, is perpendicular to the tangent).

Rice. 8. Illustration for the derivation of the centripetal acceleration formula

Section AB is move(). We are considering uniform circular motion, so:

We substitute the resulting expression for AB into the triangle similarity formula:

The concepts of "linear speed", "acceleration", "coordinate" are not enough to describe the movement along a curved trajectory. Therefore, it is necessary to introduce quantities characterizing the rotational motion.

1. The rotation period (T ) is called the time of one complete revolution. It is measured in SI units in seconds.

Examples of periods: The Earth rotates around its axis in 24 hours (), and around the Sun - in 1 year ().

Formula for calculating the period:

where is the total rotation time; - number of revolutions.

2. Rotation frequency (n ) - the number of revolutions that the body makes per unit of time. It is measured in SI units in reciprocal seconds.

Formula for finding the frequency:

where is the total rotation time; - number of revolutions

Frequency and period are inversely proportional:

3. angular velocity () called the ratio of the change in the angle at which the body turned to the time during which this turn occurred. It is measured in SI units in radians divided by seconds.

Formula for finding the angular velocity:

where is the change in angle; is the time it took for the turn to take place.

Basic Laws of Dynamics. Newton's laws - first, second, third. Galileo's principle of relativity. The law of universal gravitation. The force of gravity. Forces of elasticity. Weight. Friction forces - rest, sliding, rolling + friction in liquids and gases.

Kinematics. Basic concepts. Uniform rectilinear motion. Uniform movement. Uniform circular motion. Reference system. Trajectory, displacement, path, equation of motion, speed, acceleration, relationship between linear and angular velocity.

simple mechanisms. Lever (lever of the first kind and lever of the second kind). Block (fixed block and movable block). Inclined plane. Hydraulic Press. The golden rule of mechanics

Conservation laws in mechanics. Mechanical work, power, energy, law of conservation of momentum, law of conservation of energy, equilibrium of solids

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Mechanical vibrations. Free and forced vibrations. Harmonic vibrations. Elastic oscillations. Mathematical pendulum. Energy transformations during harmonic vibrations

mechanical waves. Velocity and wavelength. Traveling wave equation. Wave phenomena (diffraction, interference...)

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Among the various types of curvilinear motion, of particular interest is uniform motion of a body in a circle. This is the simplest form of curvilinear motion. At the same time, any complex curvilinear motion of a body in a sufficiently small section of its trajectory can be approximately considered as uniform motion along a circle.

Such a movement is made by points of rotating wheels, turbine rotors, artificial satellites rotating in orbits, etc. With uniform motion in a circle, the numerical value of the speed remains constant. However, the direction of the velocity during such a movement is constantly changing.

The speed of the body at any point of the curvilinear trajectory is directed tangentially to the trajectory at this point. This can be seen by observing the work of a disc-shaped grindstone: pressing the end of a steel rod to a rotating stone, you can see hot particles coming off the stone. These particles fly at the same speed that they had at the moment of separation from the stone. The direction of the sparks always coincides with the tangent to the circle at the point where the rod touches the stone. Sprays from the wheels of a skidding car also move tangentially to the circle.

Thus, the instantaneous velocity of the body at different points of the curvilinear trajectory has different directions, while the modulus of velocity can either be the same everywhere or change from point to point. But even if the modulus of speed does not change, it still cannot be considered constant. After all, speed is a vector quantity, and for vector quantities, the modulus and direction are equally important. That's why curvilinear motion is always accelerated, even if the modulus of speed is constant.

Curvilinear motion can change the speed modulus and its direction. Curvilinear motion, in which the modulus of speed remains constant, is called uniform curvilinear motion. Acceleration during such movement is associated only with a change in the direction of the velocity vector.

Both the modulus and the direction of acceleration must depend on the shape of the curved trajectory. However, it is not necessary to consider each of its myriad forms. Representing each section as a separate circle with a certain radius, the problem of finding acceleration in a curvilinear uniform motion will be reduced to finding acceleration in a body moving uniformly along a circle.

Uniform motion in a circle is characterized by a period and frequency of circulation.

The time it takes for a body to make one revolution is called circulation period.

With uniform motion in a circle, the period of revolution is determined by dividing the distance traveled, i.e., the circumference of the circle by the speed of movement:

The reciprocal of a period is called circulation frequency, denoted by the letter ν . Number of revolutions per unit time ν called circulation frequency:

Due to the continuous change in the direction of speed, a body moving in a circle has an acceleration that characterizes the speed of change in its direction, the numerical value of the speed in this case does not change.

With a uniform motion of a body along a circle, the acceleration at any point in it is always directed perpendicular to the speed of movement along the radius of the circle to its center and is called centripetal acceleration.

To find its value, consider the ratio of the change in the velocity vector to the time interval during which this change occurred. Since the angle is very small, we have

Topics of the USE codifier: movement in a circle with a constant modulo speed, centripetal acceleration.

Uniform circular motion is a fairly simple example of motion with an acceleration vector that depends on time.

Let the point rotate on a circle of radius . The speed of a point is constant modulo and equal to . The speed is called linear speed points.

Period of circulation is the time for one complete revolution. For the period, we have an obvious formula:

. (1)

Frequency of circulation is the reciprocal of the period:

The frequency indicates how many complete revolutions the point makes per second. The frequency is measured in rpm (revolutions per second).

Let, for example, . This means that during the time the point makes one complete
turnover. The frequency in this case is equal to: about / s; The point makes 10 complete revolutions per second.

Angular velocity.

Consider the uniform rotation of a point in the Cartesian coordinate system. Let's place the origin of coordinates in the center of the circle (Fig. 1).

Rice. 1. Uniform circular motion

Let be the initial position of the point; in other words, for , the point had coordinates . Let the point turn through an angle in time and take the position .

The ratio of the angle of rotation to time is called angular velocity point rotation:

. (2)

Angle is usually measured in radians, so angular velocity is measured in rad/s. For a time equal to the period of rotation, the point rotates through an angle. That's why

. (3)

Comparing formulas (1) and (3), we obtain the relationship between linear and angular velocities:

. (4)

The law of motion.

Let us now find the dependence of the coordinates of the rotating point on time. We see from Fig. 1 that

But from formula (2) we have: . Consequently,

. (5)

Formulas (5) are the solution to the main problem of mechanics for the uniform motion of a point along a circle.

centripetal acceleration.

Now we are interested in the acceleration of the rotating point. It can be found by differentiating relations (5) twice:

Taking into account formulas (5), we have:

(6)

The resulting formulas (6) can be written as a single vector equality:

(7)

where is the radius vector of the rotating point.

We see that the acceleration vector is directed opposite to the radius vector, i.e., towards the center of the circle (see Fig. 1). Therefore, the acceleration of a point moving uniformly in a circle is called centripetal.

In addition, from formula (7) we obtain an expression for the modulus of centripetal acceleration:

(8)

We express the angular velocity from (4)

and substitute into (8) . Let's get one more formula for centripetal acceleration.

Since the linear speed uniformly changes direction, then the movement along the circle cannot be called uniform, it is uniformly accelerated.

Angular velocity

Pick a point on the circle 1 . Let's build a radius. For a unit of time, the point will move to the point 2 . In this case, the radius describes the angle. The angular velocity is numerically equal to the angle of rotation of the radius per unit time.

Period and frequency

Rotation period T is the time it takes the body to make one revolution.

RPM is the number of revolutions per second.

The frequency and period are related by the relation

Relationship with angular velocity

Line speed

Each point on the circle moves at some speed. This speed is called linear. The direction of the linear velocity vector always coincides with the tangent to the circle. For example, sparks from under a grinder move, repeating the direction of instantaneous speed.

Consider a point on a circle that makes one revolution, the time that is spent - this is the period T.The path that the point overcomes is the circumference of the circle.

centripetal acceleration

When moving along a circle, the acceleration vector is always perpendicular to the velocity vector, directed to the center of the circle.

Using the previous formulas, we can derive the following relations

Points lying on the same straight line emanating from the center of the circle (for example, these can be points that lie on the wheel spoke) will have the same angular velocities, period and frequency. That is, they will rotate in the same way, but with different linear speeds. The farther the point is from the center, the faster it will move.

The law of addition of velocities is also valid for rotational motion. If the motion of a body or frame of reference is not uniform, then the law applies to instantaneous velocities. For example, the speed of a person walking along the edge of a rotating carousel is equal to the vector sum of the linear speed of rotation of the edge of the carousel and the speed of the person.

The Earth participates in two main rotational movements: daily (around its axis) and orbital (around the Sun). The period of rotation of the Earth around the Sun is 1 year or 365 days. The Earth rotates around its axis from west to east, the period of this rotation is 1 day or 24 hours. Latitude is the angle between the plane of the equator and the direction from the center of the Earth to a point on its surface.

According to Newton's second law, the cause of any acceleration is a force. If a moving body experiences centripetal acceleration, then the nature of the forces that cause this acceleration may be different. For example, if a body moves in a circle on a rope tied to it, then the acting force is the elastic force.

If a body lying on a disk rotates along with the disk around its axis, then such a force is the force of friction. If the force ceases to act, then the body will continue to move in a straight line

Consider the movement of a point on a circle from A to B. The linear velocity is equal to

Now let's move on to a fixed system connected to the earth. The total acceleration of point A will remain the same both in absolute value and in direction, since the acceleration does not change when moving from one inertial frame of reference to another. From the point of view of a stationary observer, the trajectory of point A is no longer a circle, but a more complex curve (cycloid), along which the point moves unevenly.