Rectilinear and curvilinear motion. Rectilinear motion and motion along the circumference of a material point

If acceleration material point is equal to zero at all times, then the speed of its movement is constant in magnitude and in direction. The trajectory in this case is a straight line. The motion of a material point under the formulated conditions is called uniform rectilinear. With rectilinear motion, the centripetal component of acceleration is absent, and since the motion is uniform, the tangential component of acceleration is zero.

If the acceleration remains constant in time (), then the movement is called equally variable or uneven. Equally variable motion can be uniformly accelerated if a > 0, and equally slow if a< 0. В этом случае мгновенное ускорение оказывается равным среднему ускорению за любой промежуток времени. Тогда из формулы (1.5) следует а = Dv/Dt = (v-v o)/t, откуда

(1.7)

where v o - initial speed at t=0, v - speed at time t.

According to formula (1.4) ds = vdt. Then

Because for uniform motion a=const, then

(1.8)

Formulas (1.7) and (1.8) are valid not only for uniformly variable (non-uniform) rectilinear motion, but also for free fall body and for the movement of a body thrown upwards. In the last two cases, a \u003d g \u003d 9.81 m / s 2.

For uniform rectilinear motion v = v o = const, a = 0, and formula (1.8) takes the form s = vt.

Circular motion is the simplest case of curvilinear motion. The speed v of movement of a material point along a circle is called linear. With a constant modulo linear velocity, the motion in a circle is uniform. There is no tangential acceleration of a material point during uniform motion along a circle, and t \u003d 0. This means that there is no change in speed modulo. The change in the linear velocity vector in the direction is characterized by normal acceleration, and n ¹ 0. At each point of the circular trajectory, the vector a n is directed along the radius to the center of the circle.

and n \u003d v 2 / R, m / s 2. (1.9)

The resulting acceleration is indeed centripetal (normal), since at Dt->0 Dj also tends to zero (Dj->0) and the vectors and will be directed along the radius of the circle to its center.

Along with the linear speed v uniform motion a material point along a circle is characterized by an angular velocity. The angular velocity is the ratio of the angle of rotation Dj of the radius vector to the time interval during which this rotation occurred,

Rad/s (1.10)

For uneven motion, the concept of instantaneous angular velocity is used

.

The time interval t, during which the material point makes one complete revolution around the circumference, is called the rotation period, and the reciprocal of the period is the rotation frequency: n \u003d 1 / T, s -1.

For one period, the angle of rotation of the radius vector of a material point is 2π rad, therefore, Dt \u003d T, whence the rotation period, and the angular velocity is a function of the period or frequency of rotation

It is known that with a uniform motion of a material point along a circle, the path traveled by it depends on the time of movement and linear speed: s = vt, m. The path that a material point passes along a circle with radius R, for a period, is equal to 2πR. The time required for this is equal to the period of rotation, that is, t \u003d T. And, therefore,

2πR = vT, m (1.11)

and v = 2nR/T = 2πnR, m/s. Since the angle of rotation of the radius vector of a material point during the rotation period T is equal to 2π, then, based on (1.10), with Dt = T, . Substituting into (1.11), we obtain and from here we find the relationship between the linear and angular velocity

Angular velocity is a vector quantity. The angular velocity vector is directed from the center of the circle along which the material point moves with linear velocity v, perpendicular to the plane of the circle according to the rule of the right screw.

At uneven movement of a material point along a circle, the linear and angular velocities change. By analogy with linear acceleration in this case, the concept of average angular acceleration and instantaneous is introduced: . The relation between tangential and angular accelerations has the form .

With the help of this lesson, you will be able to independently study the topic “Rectilinear and curvilinear motion. The motion of a body in a circle with a constant modulo velocity. First, we characterize rectilinear and curvilinear motion by considering how, in these types of motion, the velocity vector and the force applied to the body are related. Next, consider special case when the body moves in a circle with a constant modulo speed.

In the previous lesson, we looked at issues related to the law gravity. The topic of today's lesson is closely related to this law, we will turn to the uniform motion of a body in a circle.

Earlier we said that motion - this is a change in the position of a body in space relative to other bodies over time. Movement and direction of movement are characterized, among other things, by speed. The change in speed and the type of movement itself are associated with the action of a force. If a force acts on a body, then the body changes its speed.

If the force is directed parallel to the movement of the body, then such a movement will be straightforward(Fig. 1).

Rice. one. Rectilinear motion

curvilinear there will be such a movement when the speed of the body and the force applied to this body are directed relative to each other at a certain angle (Fig. 2). In this case, the speed will change its direction.

Rice. 2. Curvilinear motion

So, at rectilinear motion the velocity vector is directed in the same direction as the force applied to the body. BUT curvilinear motion is such a movement when the velocity vector and the force applied to the body are located at some angle to each other.

Consider a special case of curvilinear motion, when the body moves in a circle with a constant speed in absolute value. When a body moves in a circle at a constant speed, only the direction of the speed changes. Modulo it remains constant, but the direction of the velocity changes. Such a change in speed leads to the presence of an acceleration in the body, which is called centripetal.

Rice. 6. Movement along a curved path

If the trajectory of the body's motion is a curve, then it can be represented as a set of motions along arcs of circles, as shown in Fig. 6.

On fig. 7 shows how the direction of the velocity vector changes. The speed during such a movement is directed tangentially to the circle along the arc of which the body moves. Thus, its direction is constantly changing. Even if the modulo speed remains constant, a change in speed leads to an acceleration:

In this case acceleration will be directed towards the center of the circle. That is why it is called centripetal.

Why is centripetal acceleration directed towards the center?

Recall that if a body moves along a curved path, then its velocity is tangential. Velocity is a vector quantity. A vector has a numerical value and a direction. The speed as the body moves continuously changes its direction. That is, the difference in speeds at different points in time will not be equal to zero (), in contrast to a rectilinear uniform motion.

So, we have a change in speed over a certain period of time. Relation to is acceleration. We come to the conclusion that, even if the speed does not change in absolute value, a body that performs uniform motion in a circle has an acceleration.

Where is this acceleration directed? Consider Fig. 3. Some body moves curvilinearly (in an arc). The speed of the body at points 1 and 2 is tangential. The body moves uniformly, that is, the modules of the velocities are equal: , but the directions of the velocities do not coincide.

Rice. 3. Movement of the body in a circle

Subtract the speed from and get the vector . To do this, you need to connect the beginnings of both vectors. In parallel, we move the vector to the beginning of the vector . We build up to a triangle. The third side of the triangle will be the velocity difference vector (Fig. 4).

Rice. 4. Velocity difference vector

The vector is directed towards the circle.

Consider a triangle formed by the velocity vectors and the difference vector (Fig. 5).

Rice. 5. Triangle formed by velocity vectors

This triangle is isosceles (velocity modules are equal). So the angles at the base are equal. Let's write the equation for the sum of the angles of a triangle:

Find out where the acceleration is directed at a given point of the trajectory. To do this, we begin to bring point 2 closer to point 1. With such an unlimited diligence, the angle will tend to 0, and the angle - to. The angle between the velocity change vector and the velocity vector itself is . The speed is directed tangentially, and the velocity change vector is directed towards the center of the circle. This means that the acceleration is also directed towards the center of the circle. That is why this acceleration is called centripetal.

How to find centripetal acceleration?

Consider the trajectory along which the body moves. In this case, this is an arc of a circle (Fig. 8).

Rice. 8. Movement of the body in a circle

The figure shows two triangles: a triangle formed by the velocities, and a triangle formed by the radii and the displacement vector. If points 1 and 2 are very close, then the displacement vector will be the same as the path vector. Both triangles are isosceles with the same vertex angles. So the triangles are similar. This means that the corresponding sides of the triangles are in the same ratio:

The displacement is equal to the product of speed and time: . Substituting this formula, you can get the following expression for centripetal acceleration:

Angular velocity denoted Greek letter omega (ω), it tells about the angle through which the body rotates per unit time (Fig. 9). This is the magnitude of the arc, in degrees, traversed by the body in some time.

Rice. 9. Angular speed

Let us note that if solid rotates, then the angular velocity for any points on this body will be a constant value. The point is closer to the center of rotation or farther - it does not matter, that is, it does not depend on the radius.

The unit of measurement in this case will be either degrees per second (), or radians per second (). Often the word "radian" is not written, but simply written. For example, let's find what the angular velocity of the Earth is. The earth makes a full rotation in one hour, and in this case we can say that the angular velocity is equal to:

Also pay attention to the relationship between angular and linear velocities:

The linear speed is directly proportional to the radius. The larger the radius, the greater the linear speed. Thus, moving away from the center of rotation, we increase our linear speed.

It should be noted that motion in a circle at a constant speed is a special case of motion. However, circular motion can also be uneven. The speed can change not only in direction and remain the same in absolute value, but also change in its value, i.e., in addition to changing direction, there is also a change in the speed module. In this case, we are talking about the so-called accelerated circular motion.

What is a radian?

There are two units for measuring angles: degrees and radians. In physics, as a rule, the radian measure of an angle is the main one.

Let's construct a central angle , which relies on an arc of length .

Movement is a change of position
bodies in space relative to others
bodies over time. Movement and
the direction of movement is characterized in
including speed. Change
speed and the type of movement itself are associated with
the action of force. If the body is affected
force, the body changes its speed.

If the force is parallel
movement of the body, in one direction, then this
movement will be straight.

Such a movement will be curvilinear,
when the speed of the body and the force applied to
this body are directed relative to each other
friend at some angle. In this case
speed will change
direction.

So, for a rectilinear
movement, the velocity vector is directed to that
same side as the force applied to
body. And curvilinear
movement is the movement
when the velocity vector and force,
attached to the body, located under
some angle to each other.

centripetal acceleration

CENTRIPEAL
ACCELERATION
Consider a special case
curvilinear motion when the body
moves in a circle with constant
speed module. When the body moves
in a circle at a constant speed, then
only the direction of speed changes. By
modulo, it remains constant, and
the direction of the speed changes. Such
change in speed leads to
body of acceleration, which
called centripetal.

If the trajectory of the body is
curve, it can be represented as
set of movements along arcs
circles, as shown in Fig.
3.

On fig. 4 shows how the direction changes
velocity vector. The speed of this movement
directed tangentially to the circle, along the arc
which the body is moving. Thus, her
direction is constantly changing. Even
modulo speed remains constant,
change in speed leads to the appearance of acceleration:

In this case, the acceleration will be
directed towards the center of the circle. So
it is called centripetal.
It can be calculated using the following
formula:

Angular velocity. relationship between angular and linear velocities

ANGULAR VELOCITY. CONNECTION
CORNER AND LINE
SPEEDS
Some characteristics of the movement
circles
Angular velocity is denoted by the Greek
with the letter omega (w), it indicates which
angle rotates the body per unit time.
This is the magnitude of the arc in degrees,
passed by the body in some time.
Note that if a rigid body rotates, then
angular velocity for any points on this body
will be a constant value. closer point
is located towards the center of rotation or farther -
it doesn't matter, i.e. does not depend on the radius.

The unit of measure in this case would be
either degrees per second or radians
give me a sec. Often the word "radian" is not written, but
just write c-1. For example, let's find
what is the angular velocity of the earth. Earth
makes a full 360° turn in 24 hours, and
In this case, one can say that
angular velocity is equal.

Also note the relationship of angular
speed and line speed:
V = w. R.
It should be noted that the movement
circles with constant speed is a quotient
movement case. However, circular motion
may also be uneven. speed can
change not only in direction and remain
identical in modulus, but also change in its own way
meaning, i.e., apart from changing direction,
there is also a change in the modulus of speed. AT
In this case, we are talking about the so-called
accelerated circular motion.

Depending on the shape of the trajectory, the movement can be divided into rectilinear and curvilinear. Most often, you will encounter curvilinear movements when the path is represented as a curve. An example of this type of movement 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 displacement in curvilinear motion

Definition 1

Curvilinear motion called the movement, the trajectory of which is a curved line. If the 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 velocity of the body is tangential at the same point of the trajectory, where in this moment a moving object is located, as shown in Figure 2.

Figure 2. Instantaneous speed in curvilinear motion

Definition 2

Curvilinear motion of a material point called uniform when the modulus of speed is constant (motion in a circle), and uniformly accelerated with a changing direction and modulus of speed (movement of a thrown body).

Curvilinear motion is always accelerated. This is explained by the fact that even with an unchanged speed modulus, but a changed direction, there is always an acceleration.

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

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

Figure 3. Splitting curvilinear motion into translational

Now for each section, you can apply the law of rectilinear motion. This principle is accepted.

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

Figure 4. Partitioning of a curvilinear motion into motions along arcs of circles

Remark 1

To record a curvilinear movement, it is necessary to be able to describe movement along a circle, to represent an arbitrary movement in the form of 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 you to determine all the characteristics of the motion from the available initial conditions.

Example 1

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

Decision

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

s → = R 1 + 2 R 2 + R 3 .

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

l ~ A B \u003d π R 1 + R 2 + R 3.

Answer: s → \u003d R 1 + 2 R 2 + R 3, l ~ A B \u003d π R 1 + R 2 + R 3.

Example 2

The dependence of the path traveled by the body on time is given, represented by the equation s (t) \u003d A + B t + C t 2 + D t 3 (C \u003d 0, 1 m / s 2, D \u003d 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

Decision

Answer: t = 60 s.

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