Circular motion speed graph. Movement of a material point along a circle
Uniform circular motion- this is the simplest example. For example, the end of the clock hand moves along the dial along the circle. The speed of a body in a circle is called line speed.
With a uniform motion of the body along a circle, the modulus of the velocity of the body does not change over time, that is, v = const, but only the direction of the velocity vector changes in this case (a r = 0), and the change in the velocity vector in the direction is characterized by a value called centripetal acceleration() a n or a CA. At each point, the centripetal acceleration vector is directed to the center of the circle along the radius.
The module of centripetal acceleration is equal to
a CS \u003d v 2 / R
Where v is the linear speed, R is the radius of the circle
Rice. 1.22. The movement of the body in a circle.
When describing the motion of a body in a circle, use radius turning angle is the angle φ by which the radius drawn from the center of the circle to the point where the moving body is at that moment rotates in time t. The rotation angle is measured in radians. equal to the angle between two radii of a circle, the length of the arc between which is equal to the radius of the circle (Fig. 1.23). That is, if l = R, then
1 radian= l / R
Because circumference is equal to
l = 2πR
360 o \u003d 2πR / R \u003d 2π rad.
Consequently
1 rad. \u003d 57.2958 about \u003d 57 about 18 '
Angular velocity uniform motion of the body in a circle is the value ω, equal to the ratio of the angle of rotation of the radius φ to the time interval during which this rotation is made:
ω = φ / t
The unit of measure for angular velocity is radians per second [rad/s]. The linear velocity modulus is determined by the ratio of the distance traveled l to the time interval t:
v= l / t
Line speed with uniform motion along a circle, it is directed tangentially at a given point on the circle. When the point moves, the length l of the circular arc traversed by the point is related to the angle of rotation φ by the expression
l = Rφ
where R is the radius of the circle.
Then, in the case of uniform motion of the point, the linear and angular velocities are related by the relation:
v = l / t = Rφ / t = Rω or v = Rω
Rice. 1.23. Radian.
Period of circulation- this is the period of time T, during which the body (point) makes one revolution around the circumference. Frequency of circulation- this is the reciprocal of the circulation period - the number of revolutions per unit time (per second). The frequency of circulation is denoted by the letter n.
n=1/T
For one period, the angle of rotation φ of the point is 2π rad, therefore 2π = ωT, whence
T = 2π / ω
That is, the angular velocity is
ω = 2π / T = 2πn
centripetal acceleration can be expressed in terms of the period T and the frequency of revolution n:
a CS = (4π 2 R) / T 2 = 4π 2 Rn 2
Circular motion is a special case of curvilinear motion. The speed of the body at any point of the curvilinear trajectory is directed tangentially to it (Fig. 2.1). In this case, the speed as a vector can change both in absolute value (value) and in direction. If the speed module remains unchanged, then one speaks of uniform curvilinear motion.
Let the body move in a circle with a constant velocity from point 1 to point 2.
In this case, the body will cover a path equal to the length of the arc ℓ 12 between points 1 and 2 in time t. During the same time t, the radius-vector R drawn from the center of the circle 0 to the point will rotate through the angle Δφ.
The velocity vector at point 2 differs from the velocity vector at point 1 by direction by ΔV:
;
To characterize the change in the velocity vector by δv, we introduce the acceleration:
(2.4)
Vector at any point of the trajectory is directed along the radius Rk center circle perpendicular to the velocity vector V 2 . Therefore, the acceleration , which characterizes the change in speed during curvilinear motion in direction, called centripetal or normal. Thus, the movement of a point along a circle with a constant modulo speed is accelerated.
If the speed changes not only in direction, but also in absolute value (value), then in addition to normal acceleration also introduce tangent (tangential) acceleration , which characterizes the change in speed in magnitude:
or
Directed vector tangentially at any point of the trajectory (i.e. coincides with the direction of the vector ). Angle between vectors and is equal to 90 0 .
The total acceleration of a point moving along a curved path is defined as a vector sum (Fig. 2.1.).
.
Vector modulus
.
Angular Velocity and Angular Acceleration
When moving a material point around the circumference the radius-vector R, drawn from the center of the circle O to the point, rotates through the angle Δφ (Fig. 2.1). To characterize rotation, the concepts of angular velocity ω and angular acceleration ε are introduced.
The angle φ can be measured in radians. 1 rad is equal to the angle that rests on the arc ℓ equal to the radius R of the circle, i.e.
or ℓ 12 = Rφ (2.5.)
We differentiate equation (2.5.)
(2.6.)
Value dℓ/dt=V inst. The value ω \u003d dφ / dt is called angular velocity(measured in rad/s). We get the relationship between linear and angular velocities:
The quantity ω is vector. vector direction determined screw (gimlet) rule: it coincides with the direction of movement of the screw, oriented along the axis of rotation of the point or body and rotated in the direction of rotation of the body (Fig. 2.2), i.e.
.
angular accelerationcalled the vector quantity derivative of the angular velocity (instantaneous angular acceleration)
, (2.8.)
Vector coincides with the axis of rotation and is directed in the same direction as the vector , if the rotation is accelerated, and in the opposite direction, if the rotation is slow.
Speednbody per unit time is calledspeed .
The time T of one complete rotation of the body is calledrotation period . WhereinRdescribes the angle Δφ=2π radians
With that said
, (2.9)
Equation (2.8) can be written as follows:
(2.10)
Then the tangential component of the acceleration
and =R(2.11)
Normal acceleration a n can be expressed as follows:
in view of (2.7) and (2.9)
(2.12)
Then the full acceleration
For rotational motion with constant angular acceleration , the kinematics equation can be written by analogy with equation (2.1) - (2.3) for translational motion:
,
.
Rotational motion around a fixed axis is another special case of motion solid body.
Rotational motion of a rigid body around a fixed axis
its movement is called, in which all points of the body describe circles, the centers of which are on one straight line, called the axis of rotation, while the planes to which these circles belong are perpendicular axes of rotation
(fig.2.4).
In technology, this type of movement is very common: for example, the rotation of the shafts of engines and generators, turbines and aircraft propellers.
Angular velocity
. Each point of a body rotating around an axis passing through a point O, moves in a circle, and different points travel different paths in time. So, , therefore, the modulus of the speed of the point BUT more than dot AT (fig.2.5). But the radii of the circles rotate in time by the same angle. Angle - the angle between the axis OH and the radius vector , which determines the position of the point A (see Fig.2.5).
Let the body rotate uniformly, i.e., rotate through the same angles for any equal time intervals. The speed of rotation of the body depends on the angle of rotation of the radius vector, which determines the position of one of the points of the rigid body for a given period of time; it is characterized angular velocity
.
For example, if one body rotates by an angle every second, and the other by an angle, then we say that the first body rotates 2 times faster than the second.
The angular velocity of the body with uniform rotation
is called a value equal to the ratio of the angle of rotation of the body to the time interval for which this rotation occurred.
We will denote the angular velocity by the Greek letter ω
(omega). Then by definition
Angular velocity is expressed in radians per second (rad/s).
For example, the angular velocity of the Earth's rotation around its axis is 0.0000727 rad/s, and that of a grinding wheel is about 140 rad/s 1 .
The angular velocity can be expressed in terms of rotational speed
, i.e., the number of complete revolutions in 1s. If the body makes (the Greek letter "nu") revolutions in 1 s, then the time of one revolution is equal to seconds. This time is called rotation period
and denoted by the letter T. Thus, the relationship between frequency and rotation period can be represented as:
The full rotation of the body corresponds to the angle . Therefore, according to formula (2.1)
If, with uniform rotation, the angular velocity is known and at the initial moment of time the angle of rotation , then the angle of rotation of the body during the time t according to equation (2.1) is equal to:
If , then , or .
The angular velocity takes on positive values if the angle between the radius vector, which determines the position of one of the points of the rigid body, and the axis OH increases, and negative when it decreases.
Thus, we can describe the position of the points of a rotating body at any time.
Relationship between linear and angular speeds.
The speed of a point moving in a circle is often called linear speed
to emphasize its difference from angular velocity.
We have already noted that when a rigid body rotates, its different points have unequal linear velocities, but the angular velocity for all points is the same.
There is a connection between the linear velocity of any point of a rotating body and its angular velocity. Let's install it. A point on a circle with radius R, for one revolution will cover the path . Since the time of one revolution of the body is the period T, then the module of the linear velocity of the point can be found as follows:
Among various kinds curvilinear motion is of particular interest 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
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 relationship
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 traveled by a point is the circumference of a 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 is involved in two main rotational movements: diurnal (around its own 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 v A and v B respectively. Acceleration is the change in speed per unit of time. Let's find the difference of vectors.