Centripetal acceleration- component of the acceleration of a point, characterizing the speed of change in the direction of the velocity vector for a trajectory with curvature (the second component, tangential acceleration, characterizes the change in the velocity module). Directed towards the center of curvature of the trajectory, which is where the term comes from. The value is equal to the square of the speed divided by the radius of curvature. The term "centripetal acceleration" is equivalent to the term " normal acceleration" That component of the sum of forces that causes this acceleration is called centripetal force.
The simplest example of centripetal acceleration is the acceleration vector during uniform motion in a circle (directed towards the center of the circle).
Rapid acceleration in projection onto a plane perpendicular to the axis, it appears as centripetal.
Encyclopedic YouTube
-
1 / 5
A n = v 2 R (\displaystyle a_(n)=(\frac (v^(2))(R))\ ) a n = ω 2 R , (\displaystyle a_(n)=\omega ^(2)R\ ,)
Where a n (\displaystyle a_(n)\ )- normal (centripetal) acceleration, v (\displaystyle v\ )- (instantaneous) linear speed of movement along the trajectory, ω (\displaystyle \omega \ )- (instantaneous) angular velocity of this movement relative to the center of curvature of the trajectory, R (\displaystyle R\ )- radius of curvature of the trajectory at a given point. (The connection between the first formula and the second is obvious, given v = ω R (\displaystyle v=\omega R\ )).
The expressions above include absolute values. They can be easily written in vector form by multiplying by e R (\displaystyle \mathbf (e)_(R))- unit vector from the center of curvature of the trajectory to its given point:
a n = v 2 R e R = v 2 R 2 R (\displaystyle \mathbf (a) _(n)=(\frac (v^(2))(R))\mathbf (e) _(R)= (\frac (v^(2))(R^(2)))\mathbf (R) ) a n = ω 2 R . (\displaystyle \mathbf (a) _(n)=\omega ^(2)\mathbf (R) .)These formulas are equally applicable to the case of motion with a constant (in absolute value) speed and to an arbitrary case. However, in the second, one must keep in mind that centripetal acceleration is not the full acceleration vector, but only its component perpendicular to the trajectory (or, what is the same, perpendicular to the instantaneous velocity vector); the full acceleration vector then also includes a tangential component ( tangential acceleration) a τ = d v / d t (\displaystyle a_(\tau )=dv/dt\ ), in direction coinciding with the tangent to the trajectory (or, what is the same, with the instantaneous speed).
Motivation and conclusion
The fact that the decomposition of the acceleration vector into components - one along the tangent to the vector trajectory (tangential acceleration) and the other orthogonal to it (normal acceleration) - can be convenient and useful is quite obvious in itself. When moving with a constant modulus speed, the tangential component becomes equal to zero, that is, in this important particular case it remains only normal component. In addition, as can be seen below, each of these components has clearly defined properties and structure, and normal acceleration contains quite important and non-trivial geometric content in the structure of its formula. Not to mention the important special case of circular motion.
Formal conclusion
The decomposition of acceleration into tangential and normal components (the second of which is centripetal or normal acceleration) can be found by differentiating with respect to time the velocity vector, presented in the form v = v e τ (\displaystyle \mathbf (v) =v\,\mathbf (e) _(\tau )) through the unit tangent vector e τ (\displaystyle \mathbf (e)_(\tau )):
a = d v d t = d (v e τ) d t = d v d t e τ + v d e τ d t = d v d t e τ + v d e τ d l d l d t = d v d t e τ + v 2 R e n , (\displaystyle \mathbf (a) =(\frac (d\mathbf ( v) )(dt))=(\frac (d(v\mathbf (e) _(\tau )))(dt))=(\frac (\mathrm (d) v)(\mathrm (d) t ))\mathbf (e) _(\tau )+v(\frac (d\mathbf (e) _(\tau ))(dt))=(\frac (\mathrm (d) v)(\mathrm ( d) t))\mathbf (e) _(\tau )+v(\frac (d\mathbf (e) _(\tau ))(dl))(\frac (dl)(dt))=(\ frac (\mathrm (d) v)(\mathrm (d) t))\mathbf (e) _(\tau )+(\frac (v^(2))(R))\mathbf (e) _( n)\ ,)Here we use the notation for the unit vector normal to the trajectory and l (\displaystyle l\ )- for the current trajectory length ( l = l (t) (\displaystyle l=l(t)\ )); the last transition also uses the obvious
d l / d t = v (\displaystyle dl/dt=v\ )and, from geometric considerations,
d e τ d l = e n R . (\displaystyle (\frac (d\mathbf (e) _(\tau ))(dl))=(\frac (\mathbf (e) _(n))(R)).) v 2 R e n (\displaystyle (\frac (v^(2))(R))\mathbf (e) _(n)\ )Normal (centripetal) acceleration. Moreover, its meaning, the meaning of the objects included in it, as well as proof of the fact that it is indeed orthogonal to the tangent vector (that is, that e n (\displaystyle \mathbf (e)_(n)\ )- really a normal vector) - will follow from geometric considerations (however, the fact that the derivative of any vector of constant length with respect to time is perpendicular to this vector itself is a fairly simple fact; in this case we apply this statement to d e τ d t (\displaystyle (\frac (d\mathbf (e) _(\tau ))(dt)))
Notes
It is easy to notice that the absolute value of the tangential acceleration depends only on the ground acceleration, coinciding with its absolute value, in contrast to the absolute value of the normal acceleration, which does not depend on the ground acceleration, but depends on the ground speed.
The methods presented here, or variations thereof, can be used to introduce concepts such as the curvature of a curve and the radius of curvature of a curve (since in the case where the curve is a circle, R (\displaystyle R) coincides with the radius of such a circle; it is also not too difficult to show that the circle is in the plane e τ , e n (\displaystyle \mathbf (e) _(\tau ),\,e_(n)) with center in direction e n (\displaystyle e_(n)\ ) from a given point at a distance R (\displaystyle R) from it - will coincide with the given curve - trajectory - up to the second order of smallness in the distance to the given point).
Story
The first to obtain correct formulas for centripetal acceleration (or centrifugal force) was, apparently, Huygens. Almost from this time on, consideration of centripetal acceleration has become part of the usual technique for solving mechanical problems, etc.
Somewhat later, these formulas played a significant role in the discovery of the law of universal gravitation (the formula of centripetal acceleration was used to obtain the law of the dependence of gravitational force on the distance to the source of gravity, based on Kepler’s third law derived from observations).
By the 19th century, the consideration of centripetal acceleration had become completely routine both for pure science and for engineering applications.
Uniform circular motion is characterized by the movement of a body along a circle. In this case, only the direction of the velocity changes, and its magnitude remains constant.
In general, a body moves along a curved path, and it is difficult to describe. To simplify the description of curvilinear motion, it is divided into simpler types of motion. In particular, one of these types is uniform movement in a circle. Any curved trajectory of movement can be divided into sections of sufficiently small size, in which the body will approximately move along an arc that is part of a circle.
When a body moves in a circle, the linear speed is directed tangentially. Consequently, even if a body moves along an arc with a constant absolute speed, the direction of movement at each point will be different. Thus, any movement in a circle is a movement with acceleration.
Imagine a circle along which a material point moves. At the zero moment of time, it is in position A. After a certain time interval, it ends up at point B. If we draw two radius vectors from the center of the circle to point A and point B, then a certain angle will be obtained between them. Let's call it angle phi. If, over equal periods of time, a point rotates through the same angle phi, then such motion is called uniform, and the speed is called angular.
Figure 1 - angular velocity.
Angular velocity is measured in revolutions per second. One revolution per second is when a point passes along the entire circle and returns to its original position, taking one second. This turnover is called the circulation period. The reciprocal of the rotation period is called the rotation frequency. That is, how many revolutions does the point manage to make within one second. The angle formed by two radius vectors is measured in radians. A radian is the angle between two radius vectors that cut an arc of radius length on the surface of a circle.
The speed of a point moving around a circle can also be measured in radians per second. In this case, the movement of a point by one radian per second is called speed. This speed is called angular speed. That is, how many unit angles does the radius vector manage to rotate within one second? With uniform motion in a circle, the angular velocity is constant.
To determine the acceleration of motion in a circle, we plot in the figure the velocity vectors of points A and B. The angle between these vectors is equal to the angle between the radius vectors. Since acceleration is the difference between speeds taken over a certain time interval divided by this interval. Then, using parallel translation, we will transfer the beginning of the velocity vector at point A to point B. The difference between these vectors will be the vector delta V. If we divide it by the chord connecting points A and B, provided that the distance between the points is infinitely small, then we will obtain the acceleration vector directed towards the center of the circle. Which is also called centripetal acceleration.
Since linear speed uniformly changes direction, the circular motion cannot be called uniform, it is uniformly accelerated.
Angular velocity
Let's choose a point on the circle 1 . Let's build a radius. In a unit of time, the point will move to point 2 . In this case, the radius describes the angle. Angular velocity is numerically equal to the angle of rotation of the radius per unit time.
Period and frequency
Rotation period T- this is the time during which the body makes one revolution.
Rotation frequency is the number of revolutions per second.
Frequency and period are interrelated by the relationship
Relationship with angular velocity
Linear speed
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 under a grinding machine move, repeating the direction of instantaneous speed.
Consider a point on a circle that makes one revolution, the time spent is the period T. The path that a point travels is the circumference.
Centripetal acceleration
When moving in a circle, the acceleration vector is always perpendicular to the velocity vector, directed towards the center of the circle.
Using the previous formulas, we can derive the following relationships
Points lying on the same straight line emanating from the center of the circle (for example, these could be points that lie on the spokes of a wheel) will have the same angular velocities, period and frequency. That is, they will rotate the same way, but with different linear speeds. The further a point is from the center, the faster it will move.
The law of addition of speeds 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: 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 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 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 with the disk around its axis, then such a force is the friction force. If the force stops its action, 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 speed is equal to v A And vB respectively. Acceleration is the change in speed per unit time. Let's find the difference between the vectors.
Job source: Decision 3553.-20. OGE 2016 Mathematics, I.V. Yashchenko. 36 options.
Task 18. The diagram shows the distribution of land by category in the Ural, Volga, Southern and Far Eastern federal districts. Determine from the diagram which district has the smallest share of agricultural land.
1) Ural Federal District
2) Volga Federal District
3) Southern Federal District
4) Far Eastern Federal District
Solution.
Agricultural lands are colored by a sector in the form of horizontal lines (see figure). You need to choose a district in which the area of such a sector is minimal. Analysis of the figure shows that this is the Far Eastern Federal District.
Answer: 4.
Task 19. Grandma has 20 cups: 10 with red flowers, the rest with blue. Grandmother pours tea into a randomly selected cup. Find the probability that it will be a cup with blue flowers.
Solution.
Since there are exactly 20-10 = 10 cups with blue flowers, and there are 20 cups in total, then the probability of choosing a cup with blue flowers at random will be equal to
.Answer: 0,5.
Task 20. Centripetal acceleration when moving in a circle (in m/s2) can be calculated using the formula a=w^2*R where w is the angular velocity (in s-1), and R is the radius of the circle. Using this formula, find the radius R (in meters) if the angular velocity is 7.5 s-1 and the centripetal acceleration is 337.5 m/s2.
Solution.
From the formula we express the radius of the circle, we get:
and calculate it by substituting the data , , into the formula, we have.
In nature, body movement often occurs along curved lines. Almost any curvilinear movement can be represented as a sequence of movements along circular arcs. In general, when moving in a circle, the speed of a body changes as in size, so and towards.
Uniform movement around a circle
Circular motion is called uniform if the speed remains constant.
According to Newton's third law, every action causes an equal and opposite reaction. The centripetal force with which the connection acts on the body is counteracted by an equal in magnitude and oppositely directed force with which the body acts on the connection. This power F 6 named centrifugal, since it is directed radially from the center of the circle. Centrifugal force is equal in magnitude to centripetal force:
Examples
Consider the case where an athlete rotates an object tied to the end of a string around his head. The athlete feels a force applied to the arm and pulling it outward. To hold the object on the circle, the athlete (using a thread) pulls it inward. Therefore, according to Newton’s third law, an object (again through a thread) acts on the hand with an equal and opposite force, and this is the force that the athlete’s hand feels (Fig. 3.23). The force acting on an object is the inward tension of the thread.
Another example: a “hammer” sports equipment is acted upon by a cable held by the athlete (Fig. 3.24).
Let us recall that the centrifugal force acts not on a rotating body, but on a thread. If centrifugal force acted on the body then if the thread breaks, it would fly radially away from the center, as shown in Fig. 3.25, a. However, in fact, when the thread breaks, the body begins to move tangentially (Figure 3.25, b) in the direction of the speed that it had at the moment the thread broke.
Centrifugal forces are widely used.
A centrifuge is a device designed for training and testing pilots, athletes, and astronauts. The large radius (up to 15 m) and high engine power (several MW) make it possible to create centripetal acceleration of up to 400 m/s 2 . The centrifugal force presses the bodies with a force exceeding the normal force of gravity on Earth by more than 40 times. A person can withstand a temporary overload of 20-30 times if he lies perpendicular to the direction of the centrifugal force, and 6 times if he lies along the direction of this force.
3.8. Elements of describing human movement
Human movements are complex and difficult to describe. However, in a number of cases, it is possible to identify significant points that distinguish one type of movement from another. Consider, for example, the difference between running and walking.
Elements of stepping movements when walking are shown in Fig. 3.26. In walking movements, each leg alternates between supporting and carrying. The support period includes depreciation (braking the movement of the body towards the support) and repulsion, while the transfer period includes acceleration and braking.
The sequential movements of the human body and his legs when walking are shown in Fig. 3.27.
Lines A and B provide a high-quality image of the movement of the feet during walking. The top line A refers to one leg, the bottom line B to the other. Straight sections correspond to the moments of foot support on the ground, arcuate sections correspond to the moments of movement of the feet. During a period of time (a) both feet rest on the ground; then (b)- leg A is in the air, leg B continues to lean; and then (With)- again both legs rest on the ground. The faster you walk, the shorter the intervals become. (A And With).
In Fig. Figure 3.28 shows the sequential movements of the human body when running and a graphical representation of the movements of the feet. As you can see in the figure, when running there are time intervals { b, d, /), when both legs are in the air, and there are no intervals between the legs simultaneously touching the ground. This is the difference between running and walking.
Another common type of movement is pushing off the support during various jumps. The push-off is accomplished by straightening the pushing leg and swinging movements of the arms and torso. The task of repulsion is to ensure the maximum value of the initial velocity vector of the athlete’s general center of mass and its optimal direction. In Fig. 3.29 phases are shown
\ Chapter 4
DRIVING DYNAMICSMATERIAL POINT
Dynamics is a branch of mechanics that studies the movement of a body taking into account its interaction with other bodies.
In the “Kinematics” section the concepts were introduced speed And acceleration material point. For real bodies, these concepts need clarification, since for different real body points these movement characteristics may vary. For example, a curved soccer ball not only moves forward, but also rotates. The points of a rotating body move at different speeds. For this reason, the dynamics of a material point are first considered, and then the results obtained are extended to real bodies.