6. Curvilinear movement. Angular displacement, angular velocity and acceleration of a body. Path and displacement during curvilinear movement of a body.
Curvilinear movement– this is a movement whose trajectory is a curved line (for example, a circle, ellipse, hyperbola, parabola). An example of curvilinear motion is the movement of planets, the end of a clock hand along a dial, etc. In general curvilinear speed changes in magnitude and direction.
Curvilinear motion of a material point is considered uniform motion if the module speed constant (for example, uniform motion in a circle), and uniformly accelerated if the module and direction speed changes (for example, the movement of a body thrown at an angle to the horizontal).
Rice. 1.19. Trajectory and vector of movement during curvilinear movement.
When driving along curvilinear trajectory displacement vector directed along the chord (Fig. 1.19), and l- length trajectories . The instantaneous speed of the body (that is, the speed of the body at a given point of the trajectory) is directed tangentially at the point of the trajectory where the moving body is currently located (Fig. 1.20).
Rice. 1.20. Instantaneous speed during curved motion.
Curvilinear motion is always accelerated motion. That is acceleration during curved motion is always present, even if the speed module does not change, but only the direction of speed changes. The change in speed per unit time is tangential acceleration :
or 
Where v τ ,v 0 – velocity values at the moment of time t 0 +Δt And t 0 respectively.
Tangential acceleration at a given point of the trajectory, the direction coincides with the direction of the speed of movement of the body or is opposite to it.
Normal acceleration is the change in speed in direction per unit time:
Normal acceleration directed along the radius of curvature of the trajectory (towards the axis of rotation). Normal acceleration is perpendicular to the direction of velocity.
Centripetal acceleration- This normal acceleration with uniform motion in a circle.
Total acceleration during uniform curvilinear motion of a body equals:
The movement of a body along a curved path can be approximately represented as movement along the arcs of certain circles (Fig. 1.21).
Rice. 1.21. Movement of a body during curvilinear motion.
Curvilinear movement
Curvilinear movements– 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 plane xOy projections v x And v y its speed on the axis Ox And Oy and coordinates x And y points at any time t determined by formulas
A special case of curvilinear motion is circular motion. Circular motion, even uniform, is always accelerated motion: the velocity module is always directed tangentially to the trajectory, constantly changing direction, so circular motion always occurs with centripetal acceleration where r– radius of the circle.
The acceleration vector when moving in a circle is directed towards the center of the circle and perpendicular to the velocity vector.
In curvilinear motion, acceleration can be represented as the sum of normal and tangential components:
Normal (centripetal) acceleration is directed towards the center of curvature of the trajectory and characterizes the change in speed in the direction:
v – instantaneous speed value, r– radius of curvature of the trajectory at a given point.
Tangential (tangential) acceleration is directed tangentially to the trajectory and characterizes the change in speed modulo.
The total acceleration with which a material point moves is equal to:
In addition to centripetal acceleration, the most important characteristics of uniform circular motion are the period and frequency of rotation.
Circulation period- this is the time during which the body completes one revolution .
The period is indicated by the letter T(c) and is determined by the formula:
Where t- circulation time, P- the number of revolutions completed during this time.
Frequency- this is a quantity numerically equal to the number of revolutions completed per unit of time.
Frequency is denoted by a Greek letter (nu) and is found using the formula:
The frequency is measured in 1/s.
Period and frequency are mutually inverse quantities:
If a body moves in a circle with speed v, makes one revolution, then the distance traveled by this body can be found by multiplying the speed v for the time of one revolution:
l = vT. On the other hand, this path is equal to the circumference of the circle 2π r. That's why
vT = 2π r,
Where w(s -1) - angular velocity.
At a constant rotation frequency, centripetal acceleration is directly proportional to the distance from the moving particle to the center of rotation.
Angular velocity (w) – a value equal to the ratio of the angle of rotation of the radius at which the rotating point is located to the period of time during which this rotation occurred:
.
Relationship between linear and angular speeds:
The movement of a body can be considered known only when it is known how each point moves. The simplest motion of solid bodies is translational. Progressive is the motion of a rigid body in which any straight line drawn in this body moves parallel to itself.
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Movement is a change of position
bodies in space relative to others
bodies over time. Movement and
direction of movement is characterized in
including speed. Change
speed and the type of movement itself are related to
by the action of force. If the body is affected
force, then the body changes its speed.
body movement, in one direction, then this
the movement will be straight. Such a movement will be curvilinear,
when the speed of the body and the force applied to
this body, directed towards each other
friend at some angle. In this case
the speed will change
direction. So, with a straight line
motion, the speed vector is directed in that direction
the same side as the force applied to
body. And curvilinear
a movement is a movement
when the velocity vector and force,
attached to the body, located under
at some angle to each other.
Centripetal acceleration
CENTRIPTIPALACCELERATION
Let's consider a special case
curvilinear movement when the body
moves in a circle with a constant
module speed. When the body moves
around a circle at a constant speed, then
only the direction of speed changes. By
module it remains constant, but
the direction of speed changes. This
a change in speed leads to the presence of
body of acceleration, which
called centripetal. If the trajectory of the body is
curve, then it can be represented as
set of movements along arcs
circles, as shown in Fig.
3.In Fig. 4 shows how the direction changes
velocity vector. Speed during this movement
directed tangentially to a circle, along an arc
which the body moves. So her
the direction is constantly changing. Even
the absolute speed remains constant,
a change in speed leads to acceleration: IN in this case there will be acceleration
directed towards the center of the circle. That's why
it is called centripetal.
It can be calculated using the following
formula:
Angular velocity. relationship between angular and linear speeds
ANGULAR VELOCITY. CONNECTIONANGULAR AND LINEAR
SPEED
Some characteristics of the movement
circle
Angular velocity is denoted by the Greek
letter omega (w), it indicates which
the angle a body turns per unit time.
This is the magnitude of the arc in degree measure,
traveled by the body over some time.
Note if solid rotates, then
angular velocity for any points on this body
will be a constant value. Closer point
located towards the center of rotation or further –
it doesn't matter, i.e. does not depend on the radius. The unit of measurement in this case will be
either degrees per second or radians in
give me a sec. Often the word "radian" is not written, but
They simply write s-1. For example, let's find
What is the angular speed of the Earth? Earth
makes a full 360° turn in 24 hours, and in
In this case we can say that
angular velocity is equal. Also note the angular relationship
speed and linear speed:
V = w. R.
It should be noted that movement along
circles at constant speed is a particular
case of movement. However, the circular motion
may also be uneven. Speed can
change not only in direction and remain
identical in modulus, but also change in their own way
value, i.e., in addition to changing direction,
There is also a change in the speed module. IN
in this case we are talking about the so-called
accelerated movement in a circle.
We know that all bodies attract each other. In particular, the Moon, for example, is attracted to the Earth. But the question arises: if the Moon is attracted to the Earth, why does it revolve around it instead of falling towards the Earth?
In order to answer this question, it is necessary to consider the types of motion of bodies. We already know that movement can be uniform and uneven, but there are other characteristics of movement. In particular, depending on the direction, a distinction is made between straight and curvilinear movement.
Straight-line movement
It is known that a body moves under the influence of a force applied to it. You can do a simple experiment showing how the direction of movement of a body will depend on the direction of the force applied to it. To do this, you will need an arbitrary small object, a rubber cord and a horizontal or vertical support.
Ties the cord at one end to the support. At the other end of the cord we attach our object. Now, if we pull our object a certain distance and then release it, we will see how it begins to move in the direction of the support. Its movement is caused by the elastic force of the cord. This is how the Earth attracts all bodies on its surface, as well as meteorites flying from space.
Only instead of the elastic force, the force of attraction acts. Now let’s take our object with an elastic band and push it not in the direction towards/away from the support, but along it. If the object were not secured, it would simply fly away. But since it is held by a cord, the ball, moving to the side, slightly stretches the cord, which pulls it back, and the ball slightly changes its direction towards the support.
Curvilinear movement in a circle
This happens at every moment of time; as a result, the ball does not move along the original trajectory, but also not straight to the support. The ball will move around the support in a circle. The trajectory of its movement will be curvilinear. This is how the Moon moves around the Earth without falling on it.
This is how the Earth's gravity captures meteorites that fly close to the Earth, but not directly at it. These meteorites become satellites of the Earth. Moreover, how long they will stay in orbit depends on what their initial angle of motion was relative to the Earth. If their movement was perpendicular to the Earth, then they can remain in orbit indefinitely. If the angle was less than 90˚, then they will move in a descending spiral, and gradually still fall to the ground.
Circular motion with a constant modulus speed
Another point to note is that the speed of curvilinear motion around a circle varies in direction, but is the same in value. And this means that movement in a circle with a constant absolute speed occurs uniformly accelerated.
Since the direction of movement changes, it means that the movement occurs with acceleration. And since it changes equally at each moment of time, therefore, the movement will be uniformly accelerated. And the force of gravity is the force that causes constant acceleration.
The Moon moves around the Earth precisely because of this, but if suddenly the Moon’s movement ever changes, for example, a very large meteorite, then it may well leave its orbit and fall to Earth. We can only hope that this moment never comes. So it goes.
With help this lesson You can independently study the topic “Rectilinear and curvilinear motion. Movement of a body in a circle with a constant absolute speed." First, we will 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, we consider a special case when a body moves in a circle with a constant velocity in absolute value.
In the previous lesson we looked at issues related to the law universal 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.
We said earlier that movement - This is a change in the position of a body in space relative to other bodies over time. Movement and direction of movement are also characterized by speed. The change in speed and the type of movement itself are associated with the action of 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 movement will be straightforward(Fig. 1).
Rice. 1. Straight-line movement
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 movement
So, when straight motion the velocity vector is directed in the same direction as the force applied to the body. A curvilinear movement is such a movement when the velocity vector and the force applied to the body are located at a certain angle to each other.
Let us consider a special case of curvilinear motion, when a body moves in a circle with a constant velocity in absolute value. When a body moves in a circle at a constant speed, only the direction of the speed changes. In absolute value it remains constant, but the direction of the velocity changes. This change in speed leads to the presence of acceleration in the body, which is called centripetal.
Rice. 6. Movement along a curved path
If the trajectory of a body’s movement is a curve, then it can be represented as a set of movements along circular arcs, as shown in Fig. 6.
In Fig. Figure 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 absolute speed remains constant, a change in speed leads to acceleration:
In this case acceleration will be directed towards the center of the circle. That's why it's called centripetal.
Why is centripetal acceleration directed towards the center?
Recall that if a body moves along a curved path, then its speed is directed tangentially. Velocity is a vector quantity. A vector has a numerical value and a direction. The speed continuously changes its direction as the body moves. That is, the difference in speeds at different moments of time will not be equal to zero (), in contrast to rectilinear uniform motion.
So, we have a change in speed over a certain period of time. The ratio to is acceleration. We come to the conclusion that, even if the speed does not change in absolute value, a body performing uniform motion in a circle has acceleration.
Where is this acceleration directed? Let's look at Fig. 3. Some body moves curvilinearly (along an arc). The speed of the body at points 1 and 2 is directed tangentially. The body moves uniformly, that is, the velocity modules are equal: , but the directions of the velocities do not coincide.
Rice. 3. Body movement in a circle
Subtract the speed from it and get the vector. To do this, you need to connect the beginnings of both vectors. In parallel, 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.
Let's 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 (the velocity modules are equal). This means that the angles at the base are equal. Let us write down the equality for the sum of the angles of a triangle:
Let's find out where the acceleration is directed at a given point on the trajectory. To do this, we will begin to bring point 2 closer to point 1. With such unlimited diligence, the angle will tend to 0, and the angle will tend to . The angle between the velocity change vector and the velocity vector itself is . The speed is directed tangentially, and the vector of speed change 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?
Let's consider the trajectory along which the body moves. In this case it is a circular arc (Fig. 8).
Rice. 8. Body movement in a circle
The figure shows two triangles: a triangle formed by velocities, and a triangle formed by radii and displacement vector. If points 1 and 2 are very close, then the displacement vector will coincide with the path vector. Both triangles are isosceles with the same vertex angles. Thus, the triangles are similar. This means that the corresponding sides of the triangles are equally related:
The displacement is equal to the product of speed and time: . Substituting this formula, we can obtain the following expression for centripetal acceleration:
Angular velocity denoted by the Greek letter omega (ω), it indicates the angle through which the body rotates per unit time (Fig. 9). This is the magnitude of the arc in degrees passed by the body over some time.
Rice. 9. Angular velocity
Let us note that if a rigid body rotates, then the angular velocity for any points on this body will be a constant value. Whether the point is located closer to the center of rotation or further away is not important, i.e. 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 complete 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 speeds:
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 circular motion at a constant speed is a special case of motion. However, the movement around the circle may be uneven. Speed can change not only in direction and remain the same in magnitude, but also change in value, i.e., in addition to a change in direction, there is also a change in the magnitude of velocity. In this case we are talking about the so-called accelerated motion in a circle.
What is a radian?
There are two units for measuring angles: degrees and radians. In physics, as a rule, the radian measure of angle is the main one.
Let's construct a central angle that rests on an arc of length .
