The speed of movement when moving in a circle. Movement of a body in a circle with a constant modulo speed
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
Line segment 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.
An important particular case of particle motion along a given trajectory is circular motion. The position of the particle on the circle (Fig. 46) can be specified by specifying not the distance from some initial point A, but the angle formed by the radius drawn from the center O of the circle to the particle, with the radius drawn to the initial point A.
Along with the speed of movement along the trajectory, which is defined as
it is convenient to introduce the angular velocity characterizing the rate of change of the angle
The speed of movement along the trajectory is also called linear speed. Let us establish a relationship between linear and angular velocities. The length of the arc I subtending the angle is where is the radius of the circle, and the angle is measured in radians. Therefore, the angular velocity ω is also related to the linear velocity by the relation
Rice. 46. Angle sets the position of a point on a circle
Acceleration when moving along a circle, as well as during arbitrary curvilinear motion, generally has two components: tangential, directed tangentially to the circle and characterizing the speed of change in the velocity value, and normal, directed towards the center of the circle and characterizing the speed of change in the direction of speed.
The value of the normal component of acceleration, called in this case (circular motion) centripetal acceleration, is given by the general formula (3) § 8, in which the linear velocity can now be expressed in terms of angular velocity using formula (3):
Here the radius of the circle is, of course, the same for all points of the trajectory.
With uniform motion in a circle, when the value is constant, the angular velocity ω, as can be seen from (3), is also constant. In this case, it is sometimes called the cyclic frequency.
period and frequency. To characterize uniform motion in a circle, along with with it is convenient to use the period of revolution T, defined as the time during which one complete revolution is made, and the frequency - the reciprocal of the period T, which is equal to the number of revolutions per unit time:
From the definition (2) of the angular velocity follows the relationship between the quantities
This relation allows us to write formula (4) for centripetal acceleration also in the following form:
Note that angular velocity ω is measured in radians per second, and frequency is measured in revolutions per second. The dimensions with and are the same since these quantities differ only by a numerical factor
A task
Along the ring road. The rails of the toy railway form a radius ring (Fig. 47). The trailer moves along them, pushed by a rod that rotates at a constant angular velocity around a point lying inside the ring almost at the very rails. How does the speed of the trailer change as it moves?
Rice. 47. To finding the angular velocity when driving along the ring road
Solution. The angle formed by a rod with a certain direction changes with time according to a linear law: . As a direction from which the angle is measured, it is convenient to take the diameter of the circle passing through the point (Fig. 47). Point O is the center of the circle. Obviously, the central angle that determines the position of the trailer on the circle is twice the inscribed angle based on the same arc: Therefore, the angular velocity from the trailer when moving along the rails is twice the angular velocity with which the rod rotates:
Thus, the angular velocity from the trailer turned out to be constant. This means that the trailer moves along the rails evenly. Its linear speed is constant and equal to
The acceleration of the trailer with such a uniform movement in a circle is always directed towards the center O, and its module is given by expression (4):
Look at formula (4). How should it be understood: is acceleration still proportional or inversely proportional?
Explain why, with uneven motion along a circle, the angular velocity retains its meaning, but loses its meaning?
Angular velocity as a vector. In some cases, it is convenient to consider the angular velocity as a vector, the modulus of which is a constant direction perpendicular to the plane in which the circle lies. Using such a vector, one can write a formula similar to (3), which expresses the velocity vector of a particle moving in a circle.
Rice. 48. Angular Velocity Vector
We place the origin at the center O of the circle. Then, when the particle moves, its radius vector will only rotate with the angular velocity ω, and its modulus is always equal to the radius of the circle (Fig. 48). It can be seen that the velocity vector directed tangentially to the circle can be represented as the vector product of the angular velocity vector ω and the radius vector of the particle:
Vector product. By definition, the cross product of two vectors is a vector perpendicular to the plane in which the multiplied vectors lie. The choice of the vector product direction is made according to the following rule. The first multiplier is mentally turned towards the second, as if it were the handle of a wrench. The vector product is directed in the same direction as the right-handed screw would move.
If the factors in the vector product are interchanged, then it will change direction to the opposite: This means that the vector product is non-commutative.
From fig. 48 it can be seen that formula (8) will give the correct direction for the vector if the vector co is directed exactly as shown in this figure. Therefore, we can formulate the following rule: the direction of the angular velocity vector coincides with the direction of motion of a screw with a right-hand thread, the head of which turns in the same direction as the particle moves in a circle.
By definition, the module of the cross product is equal to the product of the modules of the multiplied vectors by the sine of the angle a between them:
In formula (8), the multiplied vectors w and are perpendicular to each other, therefore, as it should be in accordance with formula (3).
What can be said about the cross product of two parallel vectors?
What is the direction of the angular velocity vector of the clock hand? How do these vectors differ for the minute and hour hands?
Uniform circular motion 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 the 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
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.
Circular motion is the simplest case of curvilinear motion of a body. When a body moves around a certain 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 displacement, it is possible to calculate the length of the circular arc (path) that the body has passed.
∆ l = R ∆ φ
If the angle of rotation is small, then ∆ l ≈ ∆ s .
Let's illustrate what has been said:
Angular velocity
With curvilinear motion, the concept of angular velocity ω is introduced, that is, the rate of change in 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 (r a d s).
There is a relationship between the angular and linear velocities of the body when moving in a circle. Formula for finding the angular velocity:
With uniform motion in a circle, the speeds v and ω remain unchanged. Only the direction of the linear velocity vector changes.
In this case, a uniform movement along a circle on the body is affected by 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 by the formula:
a n = v 2 R = ω 2 R
Let us prove these relations.
Let's consider how the vector v → changes over a small period of time ∆ 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 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 \u003d R and C D \u003d ∆ v for the similar triangles considered above, 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. Assuming 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 module remains constant, and the direction of the vector changes with time, while maintaining orientation to the center of the circle. That is why this acceleration is called centripetal: the vector at any time is directed towards the center of the circle.
The record of centripetal acceleration in vector form is as follows:
a n → = - ω 2 R → .
Here R → is the radius vector of a point on a circle with origin at its center.
In the general case, acceleration when moving along a circle consists of two components - normal and tangential.
Consider the case when the body moves along the circle non-uniformly. Let us 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 \u003d v 2 - v 1 is the change in the velocity module over the interval ∆ t
The direction of full acceleration is determined by the vector sum of normal and tangential accelerations.
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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