Arc movement. Circular movement
Since the linear speed uniformly changes direction, the movement along a circle cannot be called uniform, it is equally accelerated.
Angular velocity
Choose a point on the circle 1 ... Let's build a radius. In 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 of time.
Period and frequency
Rotation period T- this is the time during which the body makes one revolution.
Rotation speed is the number of revolutions per second.
Frequency and period are interrelated by the ratio
Angular Velocity Relationship
Linear Velocity
Each point on the circle moves at a certain 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 a grinder move in the same direction as the instantaneous speed.
Consider a point on a circle that makes one revolution, the time it takes is a period T. The path that the point overcomes is the circumference.
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, the following relations can be derived
Points lying on one straight line outgoing from the center of the circle (for example, these can be points that lie on the spoke of a wheel) will have the same angular velocity, period and frequency. That is, they will rotate in the same way, but with different linear speeds. The further the point is from the center, the faster it will move.
The law of addition of velocities is also valid for rotary motion. If the movement of a body or a frame of reference is not uniform, then the law is applied for 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 person's movement speed.
The Earth participates in two main rotational movements: diurnal (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 equatorial plane and the direction from the center of the Earth to a point on its surface.
According to Newton's second law, force is the cause of any acceleration. If a moving body experiences centripetal acceleration, then the nature of the forces, the action of which is caused by this acceleration, may be different. For example, if the body moves in a circle on a rope attached to it, then acting force is the elastic force.
If a body lying on a disk rotates with the disk around its axis, then such a force is the friction force. If the force ceases to act, then the body will 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 us go over to a stationary system connected to the ground. The total acceleration of point A will remain the same both in magnitude and in direction, since when passing from one inertial frame of reference to another, the acceleration does not change. 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.
Among different types curvilinear movement is of particular interest uniform movement of the body around the circumference... This is the simplest kind of curvilinear movement. At the same time, any complex curvilinear motion of a body on a sufficiently small section of its trajectory can be approximately considered as uniform motion along a circle.
Such a movement is performed by the points of rotating wheels, turbine rotors, artificial satellites rotating in orbits, etc. uniform movement around the circumference, the numerical value of the speed remains constant. However, the direction of speed changes continuously during this movement.
The speed of movement of the body at any point of the curved trajectory is directed tangentially to the trajectory at this point. This can be seen by observing the work of a sharpener, which has the shape of a disk: pressing the end of a steel bar against a rotating stone, you can see red-hot particles coming off the stone. These particles fly with the speed that they had at the moment of separation from the stone. The direction of emission of sparks always coincides with the tangent to the circle at the point where the bar touches the stone. Spray from the wheels of the skidding car also moves tangentially to the circle.
Thus, the instantaneous velocity of the body at different points of the curved trajectory has different directions, while the modulus of velocity can be either the same everywhere, or change from point to point. But even if the speed module 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 movement is always accelerated even if the speed module is constant.
During curvilinear motion, the speed module and its direction can change. Curvilinear motion, in which the speed module remains constant, is called uniform curvilinear motion... Acceleration during such a 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 path. However, there is no need to consider each of its innumerable forms. Representing each section as a separate circle with a certain radius, the problem of finding the acceleration in curvilinear uniform motion will be reduced to finding the acceleration in the uniform motion of the body along the circumference.
Uniform circular motion is characterized by the period and frequency of revolution.
The time it takes for the body to make one revolution is called period of circulation.
With uniform movement along a circle, the period of revolution is determined by dividing the distance traveled, that is, the circumference by the speed of movement:
The reciprocal of the period is called frequency of circulation, denoted by the letter ν ... The number of revolutions per unit of time ν are called frequency of circulation:
Due to the continuous change in the direction of speed, a body moving in a circle has an acceleration, which characterizes the speed of change in its direction, the numerical value of the speed in this case does not change.
With a uniform movement of a body around a circle, the acceleration at any point 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, let us 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.
Circular motion is the simplest case of curvilinear body motion. When the body moves around some point, along with the displacement vector, it is convenient to introduce the angular displacement ∆ φ (the angle of rotation relative to the center of the circle), measured in radians.
Knowing the angular movement, you can calculate the length of the circular arc (path) that the body has traveled.
∆ l = R ∆ φ
If the angle of rotation is small, then ∆ l ≈ ∆ s.
Let us illustrate what has been said:
Angular velocity
In curvilinear motion, the concept is introduced angular velocityω, that is, the rate of change of the angle of rotation.
Definition. Angular velocity
The angular velocity at a given point of the trajectory is the limit of the ratio of the angular displacement ∆ φ to the time interval ∆ t during which it occurred. ∆ t → 0.
ω = ∆ φ ∆ t, ∆ t → 0.
The unit of measure for angular velocity is radians per second (rad s).
There is a relationship between the angular and linear velocities of a body when moving in a circle. Formula for finding the angular velocity:
With uniform motion around the circumference, the speeds v and ω remain unchanged. Only the direction of the linear velocity vector changes.
In this case, the uniform movement around the circle acts on the body centripetal, or normal acceleration directed along the radius of the circle to its center.
a n = ∆ v → ∆ t, ∆ t → 0
The centripetal acceleration module can be calculated using the formula:
a n = v 2 R = ω 2 R
Let us prove these relations.
Let us consider how the vector v → changes in a small time interval ∆ t. ∆ v → = v B → - v A →.
At points A and B, the velocity vector is directed tangentially to the circle, while the velocity modules at both points are the same.
By definition of acceleration:
a → = ∆ v → ∆ t, ∆ t → 0
Let's take a look at the picture:
Triangles OAB and BCD are similar. It follows from this that O A A B = B C C D.
If the value of the angle ∆ φ is small, the distance A B = ∆ s ≈ v ∆ t. Taking into account that O A = R and C D = ∆ v for the above similar triangles we get:
R v ∆ t = v ∆ v or ∆ v ∆ t = v 2 R
When ∆ φ → 0, the direction of the vector ∆ v → = v B → - v A → approaches the direction to the center of the circle. Taking that ∆ t → 0, we get:
a → = a n → = ∆ v → ∆ t; ∆ t → 0; a n → = v 2 R.
With uniform motion along a circle, the acceleration modulus remains constant, and the direction of the vector changes over time, maintaining the orientation to the center of the circle. That is why this acceleration is called centripetal: the vector at any time is directed to the center of the circle.
Recording centripetal acceleration in vector form looks like this:
a n → = - ω 2 R →.
Here R → is the radius vector of a point on a circle with the origin at its center.
In the general case, the acceleration when moving around a circle consists of two components - normal and tangential.
Consider the case when the body moves unevenly around the circle. Let's introduce the concept of tangential (tangential) acceleration. Its direction coincides with the direction of the linear velocity of the body and at each point of the circle is directed tangentially to it.
a τ = ∆ v τ ∆ t; ∆ t → 0
Here ∆ v τ = v 2 - v 1 is the change in the velocity modulus over the interval ∆ t
The direction of full acceleration is determined by the vector sum of the normal and tangential acceleration.
Circular motion in a plane can be described using two coordinates: x and y. At each moment of time, the speed of the body can be decomposed into components v x and v y.
If the motion is uniform, the values v x and v y as well as the corresponding coordinates will change in time according to a harmonic law with a period T = 2 π R v = 2 π ω
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In this lesson we will consider curvilinear motion, namely the uniform movement of a body along a circle. We learn what a linear speed, centripetal acceleration when a body moves in a circle. We will also introduce the values that characterize the rotational motion (rotation period, rotation frequency, angular velocity), and we will relate these values to each other.
Uniform motion along a circle means that the body rotates through the same angle for any equal period of time (see Fig. 6).
Rice. 6. Uniform circular motion
That is, the instantaneous speed module does not change:
This speed is called linear.
Although the speed module does not change, the direction of the speed changes continuously. Consider the velocity vectors at points A and B(see Fig. 7). They are directed in different directions, therefore they are not equal. If you subtract 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 the 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 the angle (α) between the velocity vectors will be close to zero to each other:
It is also known that this triangle is isosceles, so the speed modules are equal (uniform motion):
Therefore, both angles at the base of this triangle are infinitely 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 the tangent is the radius, therefore acceleration is directed along the radius to the center of the circle. Such acceleration is called centripetal.
Figure 8 shows the previously considered velocity triangle and an isosceles triangle (the two sides are the radii of the circle). These triangles are similar, since they have equal angles formed by mutually perpendicular straight lines (the radius, like the vector, are perpendicular to the tangent).
Rice. 8. Illustration for the derivation of the formula for centripetal acceleration
Section AB is displacement (). We are considering a uniform motion along a circle, therefore:
Substitute the resulting expression for AB into the triangle similarity formula:
The concepts of "linear speed", "acceleration", "coordinate" are not enough to describe movement along a curved path. Therefore, it is necessary to introduce values that characterize the rotational motion.
1. The period of rotation (T ) the time of one complete revolution is called. 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; - the number of revolutions.
2. Rotation frequency (n ) - the number of revolutions that the body makes per unit of time. Measured in SI units in reverse seconds.
Frequency formula:
where is the total rotation time; - number of revolutions
Frequency and period are inversely proportional values:
3. Angular velocity () called the ratio of the change in the angle by which the body turned, to the time during which this turn took place. Measured in SI units in radians divided by seconds.
Formula for finding the angular velocity:
where is the change in the angle; - the time during which the corner is turned.
1.Smooth circular motion
2. Angular speed of rotary movement.
3. Period of rotation.
4.Rotation frequency.
5. Connection of linear velocity with angular velocity.
6. Centripetal acceleration.
7. Equally variable movement in a circle.
8. Angular acceleration in uniform motion around a circle.
9.Tangential acceleration.
10. The law of uniformly accelerated movement in a circle.
11. Average angular velocity in uniformly accelerated motion around a circle.
12. Formulas that establish the relationship between angular velocity, angular acceleration and the angle of rotation in uniformly accelerated motion around a circle.
1.Uniform circular motion- movement in which material point in equal time intervals, equal segments of the arc of a circle pass, i.e. the point moves in a circle with a constant absolute speed. In this case, the speed is equal to the ratio of the circular arc traversed by the point to the time of movement, i.e.
and is called the linear speed of movement in a circle.
As in curvilinear motion, the velocity vector is directed tangentially to the circle in the direction of motion (Fig. 25).
2. Angular Velocity in Uniform Circular Motion- the ratio of the angle of rotation of the radius to the time of rotation:
In a uniform motion around a circle, the angular velocity is constant. In SI, angular velocity is measured in (rad / s). One radian - rad is the central angle that subtends an arc of a circle with a length equal to the radius. The total angle contains radians, i.e. in one revolution, the radius is rotated by an angle of radians.
3. Rotation period- the time interval T, during which the material point makes one complete revolution. In the SI system, the period is measured in seconds.
4. Rotation frequency- the number of revolutions made in one second. In SI units, frequency is measured in hertz (1Hz = 1). One hertz is the frequency at which one revolution is made in one second. It is easy to figure out that
If in time t the point makes n revolutions around the circle then.
Knowing the period and frequency of rotation, the angular velocity can be calculated by the formula:
5 Relationship between linear velocity and angular velocity... The length of an arc of a circle is where is the central angle, expressed in radians, that subtends the arc to the radius of the circle. Now we write the linear velocity in the form
It is often convenient to use the formulas: or The angular velocity is often referred to as the cyclic frequency, and the frequency as the linear frequency.
6. Centripetal acceleration... In uniform motion around a circle, the speed module remains unchanged, and its direction is continuously changing (Fig. 26). This means that a body moving uniformly around a circle experiences acceleration, which is directed towards the center and is called centripetal acceleration.
Suppose that a path equal to an arc of a circle has passed over a period of time. Move the vector, leaving it parallel to itself, so that its beginning coincides with the beginning of the vector at point B. The modulus of the velocity change is equal, and the modulus of centripetal acceleration is
In Fig. 26 the triangles AOB and ICE are isosceles and the angles at the vertices O and B are equal, as are the angles with mutually perpendicular sides AO and OB This means that triangles AOB and ICE are similar. Therefore, if that is, the time interval takes arbitrarily small values, then the arc can be approximately considered equal to the chord AB, i.e. ... Therefore, we can write Considering that VD =, OA = R we obtain Multiplying both sides of the last equality by, we will also obtain an expression for the centripetal acceleration modulus in uniform motion along a circle:. Considering that we get two commonly used formulas:
So, in a uniform motion around a circle, the centripetal acceleration is constant in absolute value.
It is easy to figure out that in the limit at, the angle. This means that the angles at the base of the DS of the ICE triangle tend to the value, and the velocity vector becomes perpendicular to the velocity vector, i.e. directed along the radius to the center of the circle.
7. Equally variable circular motion- movement in a circle, in which the angular velocity changes by the same amount over equal time intervals.
8. Angular acceleration in an equally variable motion along a circle- the ratio of the change in angular velocity to the time interval during which this change occurred, i.e.
where the initial value of the angular velocity, the final value of the angular velocity, the angular acceleration, in the SI system is measured in. From the last equality, we obtain formulas for calculating the angular velocity
And if .
Multiplying both sides of these equalities by and taking into account that, is the tangential acceleration, i.e. acceleration directed tangentially to the circle, we obtain the formulas for calculating the linear velocity:
And if .
9. Tangential acceleration is numerically equal to the change in speed per unit of time and is directed along the tangent to the circle. If> 0,> 0, then the motion is uniformly accelerated. If<0 и <0 – движение.
10. The law of uniformly accelerated motion in a circle... The path traversed in a circle during the time in uniformly accelerated motion is calculated by the formula:
Substituting here, canceling by, we obtain the law of uniformly accelerated motion in a circle:
Or if.
If the movement is equally slow, i.e.<0, то
11.Full acceleration in uniformly accelerated circular motion... In uniformly accelerated motion along a circle, the centripetal acceleration increases with time, because the tangential acceleration increases the linear speed. Very often, centripetal acceleration is called normal and denoted as. Since the full acceleration at the moment is determined by the Pythagorean theorem (Fig. 27).
12. Average angular velocity in uniformly accelerated motion in a circle... The average linear speed in uniformly accelerated motion in a circle is equal to. Substituting here and and reducing by we get
If, then.
12. Formulas establishing the relationship between angular velocity, angular acceleration and the angle of rotation in uniformly accelerated motion around a circle.
Substituting into the formula the quantities,,,,
and canceling by, we get
Lecture - 4. Dynamics.
1. Dynamics
2. Interaction of bodies.
3. Inertia. The principle of inertia.
4. Newton's first law.
5. Free material point.
6. Inertial frame of reference.
7. Non-inertial frame of reference.
8. Galileo's principle of relativity.
9. Transformations of Galileo.
11. Consolidation of forces.
13. Density of substances.
14. Center of mass.
15. Newton's second law.
16. Unit of measure of force.
17. Newton's third law
1. Dynamics there is a section of mechanics that studies mechanical movement, depending on the forces that cause a change in this movement.
2.Body interactions... Bodies can interact, both in direct contact, and at a distance through a special type of matter called a physical field.
For example, all bodies are attracted to each other and this attraction is carried out through the gravitational field, and the forces of attraction are called gravitational.
Bodies that carry an electric charge interact through an electric field. Electric currents interact through a magnetic field. These forces are called electromagnetic.
Elementary particles interact through nuclear fields and these forces are called nuclear.
3.Inertia... In the IV century. BC NS. the Greek philosopher Aristotle argued that the cause of the movement of a body is a force acting from another body or bodies. At the same time, according to motion, according to Aristotle, a constant force imparts a constant speed to the body, and with the cessation of the action of the force, the movement stops.
In the 16th century. Italian physicist Galileo Galilei, conducting experiments with bodies rolling along an inclined plane and with falling bodies, showed that a constant force (in this case, the weight of the body) imparts acceleration to the body.
So, on the basis of experiments, Galileo showed that force is the cause of the acceleration of bodies. Let us give Galileo's reasoning. Let a very smooth ball roll on a smooth horizontal plane. If nothing interferes with the ball, then it can roll as long as you like. If a thin layer of sand is poured on the path of the ball, then it will stop very soon, because it was acted upon by the frictional force of the sand.
So Galileo came to the formulation of the principle of inertia, according to which a material body retains a state of rest or uniform rectilinear motion, if external forces do not act on it. Often this property of matter is called inertia, and the movement of a body without external influences is called inertial motion.
4. Newton's first law... In 1687, on the basis of Galileo's principle of inertia, Newton formulated the first law of dynamics - Newton's first law:
A material point (body) is in a state of rest or uniform rectilinear motion, if other bodies do not act on it, or the forces acting from other bodies are balanced, i.e. compensated.
5.Free material point- a material point on which other bodies do not act. Sometimes they say - an isolated material point.
6. Inertial frame of reference (ISO)- a frame of reference relative to which an isolated material point moves rectilinearly and uniformly, or is at rest.
Any frame of reference that moves uniformly and rectilinearly relative to the IFR is inertial,
Let's give one more formulation of Newton's first law: There are frames of reference relative to which a free material point moves rectilinearly and uniformly, or is at rest. Such frames of reference are called inertial. Often Newton's first law is called the law of inertia.
Newton's first law can also be formulated like this: any material body resists a change in its speed. This property of matter is called inertia.
We are faced with the manifestation of this law every day in urban transport. When the bus picks up speed sharply, we are pressed against the back of the seat. When the bus slows down, then our body skids in the direction of the bus.
7. Non-inertial frame of reference - a frame of reference that moves unevenly relative to the IFR.
A body that is at rest or in uniform rectilinear motion relative to IFR. Moves unevenly relative to the non-inertial frame of reference.
Any rotating frame of reference is a non-inertial frame of reference, since in this system, the body experiences centripetal acceleration.
There are no bodies in nature and technology that could serve as ISO. For example, the Earth rotates around its axis and any body on its surface experiences centripetal acceleration. However, for rather short periods of time, the frame of reference associated with the Earth's surface in some approximation can be considered IFR.
8.Galileo's principle of relativity. ISO can be a lot of salt. Therefore, the question arises: how do the same mechanical phenomena look in different IFRs? Is it possible, using mechanical phenomena, to detect the motion of the IF, in which they are observed?
The answer to these questions is given by the principle of relativity of classical mechanics, discovered by Galileo.
The meaning of the principle of relativity of classical mechanics lies in the statement: all mechanical phenomena proceed in exactly the same way in all inertial reference frames.
This principle can be formulated as follows: all the laws of classical mechanics are expressed by the same mathematical formulas. In other words, no mechanical experiments will help us to detect the motion of the IRS. This means that an attempt to detect the movement of the IRS is meaningless.
We encountered the manifestation of the principle of relativity while traveling on trains. At the moment when our train is at the station, and the train standing on the next track slowly starts moving, then in the first moments it seems to us that our train is moving. But it also happens the other way around, when our train is gradually picking up speed, it seems to us that the movement was started by a neighboring train.
In the given example, the principle of relativity manifests itself during small intervals of time. With an increase in speed, we begin to feel the jolts of the wagon rocking, that is, our frame of reference becomes non-inertial.
So, an attempt to detect the movement of ISO is meaningless. Therefore, it is absolutely indifferent which IRF is considered to be motionless and which one is to be considered as moving.
9. Galileo transformations... Let two IFRs and move relative to each other with speed. According to the principle of relativity, we can assume that the IFR K is motionless, and the IFR moves relatively with speed. For simplicity, let us assume that the corresponding coordinate axes of the systems and are parallel, and the axes and coincide. Let at the moment of the beginning of the systems coincide and the movement occurs along the axes and, i.e. (Fig. 28)
11. The addition of forces... If two forces are applied to the particle, then the resulting force is equal to their vector force, i.e. the diagonal of the parallelogram built on the vectors and (Fig. 29).
The same rule applies to the decomposition of a given force into two component forces. To do this, on the vector of a given force, as on a diagonal, a parallelogram is built, the sides of which coincide with the direction of the constituent forces applied to a given particle.
If several forces are applied to the particle, then the resultant is equal to the geometric sum of all forces:
12.Weight... Experience has shown that the ratio of the modulus of force to the modulus of acceleration, which this force imparts to the body, is a constant value for a given body and is called the mass of the body:
From the last equality it follows that the greater the mass of the body, the greater the force must be applied to change its speed. Consequently, the greater the mass of the body, the more inert it is, i.e. mass is a measure of the inertia of bodies. The mass thus determined is called the inert mass.
In SI, mass is measured in kilograms (kg). One kilogram is the mass of distilled water in a volume of one cubic decimeter taken at a temperature
13. Density of matter- the mass of a substance contained in a unit of volume or the ratio of body mass to its volume
Density is measured in SI (). Knowing the density of the body and its volume, you can calculate its mass by the formula. Knowing the density and mass of the body, its volume is calculated by the formula.
14.Center of mass- a point of the body, which has the property that if the direction of action of the force passes through this point, the body moves translationally. If the direction of action does not pass through the center of mass, then the body moves, while rotating around its center of mass
15. Newton's second law... In IFR, the sum of the forces acting on a body is equal to the product of the body's mass by the acceleration imparted to it by this force
16.Force unit... In SI, force is measured in newtons. One newton (n) is a force that acts on a body weighing one kilogram and imparts acceleration to it. That's why .
17. Newton's third law... The forces with which two bodies act on each other are equal in magnitude, opposite in direction and act along one straight line connecting these bodies.
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