Centripetal acceleration is the acceleration component of a point characterizing the rate 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 modulus). Directed to the center of the trajectory curvature, which is the reason for the term. Equal in magnitude to the square of the velocity 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 the centripetal force.
The simplest example of centripetal acceleration is the acceleration vector for uniform motion along a circle (directed towards the center of the circle).
Blast Acceleration when projected onto a plane perpendicular to the axis, it appears as centripetal.
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A n \u003d v 2 R (\\ displaystyle a_ (n) \u003d (\\ frac (v ^ (2)) (R)) \\) a n \u003d ω 2 R, (\\ displaystyle a_ (n) \u003d \\ 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 \\) - the radius of curvature of the trajectory at a given point. (The connection between the first formula and the second is obvious, given v \u003d ω R (\\ displaystyle v \u003d \\ 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:
an \u003d v 2 R e R \u003d v 2 R 2 R (\\ displaystyle \\ mathbf (a) _ (n) \u003d (\\ frac (v ^ (2)) (R)) \\ mathbf (e) _ (R) \u003d (\\ frac (v ^ (2)) (R ^ (2))) \\ mathbf (R)) a n \u003d ω 2 R. (\\ displaystyle \\ mathbf (a) _ (n) \u003d \\ omega ^ (2) \\ mathbf (R).)These formulas are equally applicable to the case of motion with constant (in absolute value) speed, and to an arbitrary case. However, in the second one, it should be borne in mind that the centripetal acceleration is not a full vector of acceleration, but only its component perpendicular to the trajectory (or, which is the same, perpendicular to the vector of instantaneous velocity); the total acceleration vector then also includes the tangential component ( tangential acceleration) a τ \u003d d v / d t (\\ displaystyle a _ (\\ tau) \u003d dv / dt \\), in the direction coinciding with the tangent to the trajectory (or, which is the same, with the instantaneous velocity).
Motivation and withdrawal
The fact that the decomposition of the acceleration vector into components - one along the vector tangent to the trajectory (tangential acceleration) and the other orthogonal to it (normal acceleration) - can be convenient and useful is quite obvious in itself. When moving at a speed constant in absolute value, the tangential component becomes equal to zero, that is, in this important special case, it remains only normal component. In addition, as you can see below, each of these components has pronounced proper properties and structure, and the normal acceleration contains in the structure of its formula a rather important and non-trivial geometric content. Not to mention the important special case of circular motion.
Formal conclusion
The decomposition of the acceleration into tangential and normal components (the second of which is the centripetal or normal acceleration) can be found by differentiating in time the velocity vector, represented in the form v \u003d v e τ (\\ displaystyle \\ mathbf (v) \u003d v \\, \\ mathbf (e) _ (\\ tau)) through the unit tangent vector e τ (\\ displaystyle \\ mathbf (e) _ (\\ tau)):
a \u003d dvdt \u003d d (ve τ) dt \u003d dvdte τ + vde τ dt \u003d dvdte τ + vde τ dldldt \u003d dvdte τ + v 2 R en, (\\ displaystyle \\ mathbf (a) \u003d (\\ frac (d \\ mathbf ( v)) (dt)) \u003d (\\ frac (d (v \\ mathbf (e) _ (\\ tau))) (dt)) \u003d (\\ frac (\\ mathrm (d) v) (\\ mathrm (d) t )) \\ mathbf (e) _ (\\ tau) + v (\\ frac (d \\ mathbf (e) _ (\\ tau)) (dt)) \u003d (\\ frac (\\ mathrm (d) v) (\\ mathrm ( d) t)) \\ mathbf (e) _ (\\ tau) + v (\\ frac (d \\ mathbf (e) _ (\\ tau)) (dl)) (\\ frac (dl) (dt)) \u003d (\\ n) \\,)Here we used the notation for the unit normal vector to the trajectory and l (\\ displaystyle l \\) - for the current length of the trajectory ( l \u003d l (t) (\\ displaystyle l \u003d l (t) \\)); the last transition also used the obvious
d l / d t \u003d v (\\ displaystyle dl / dt \u003d v \\)and, for geometric reasons,
d e τ d l \u003d e n R. (\\ displaystyle (\\ frac (d \\ mathbf (e) _ (\\ tau)) (dl)) \u003d (\\ 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 really orthogonal to the tangent vector (that is, that e n (\\ displaystyle \\ mathbf (e) _ (n) \\) - indeed the 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)))
Remarks
It is easy to see 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 their variants 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 when 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 a circle in a plane e τ, e n (\\ displaystyle \\ mathbf (e) _ (\\ tau), \\, e_ (n)) centered towards e n (\\ displaystyle e_ (n) \\) from a given point at a distance R (\\ displaystyle R) from it - will coincide with the given curve - the trajectory - up to the second order of smallness in terms of distance to a given point).
Story
The first correct formulas for centripetal acceleration (or centrifugal force) was apparently obtained by Huygens. Practically from this time on, the consideration of centripetal acceleration has been included in 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 centripetal acceleration formula was used to obtain the law of dependence gravitational force from the distance to the source of gravity, based on the Kepler's third law derived from observations).
TO XIX century consideration of centripetal acceleration is already becoming completely routine for both pure science and engineering applications.
With uniform motion around the circumference, the body moves with centripetal acceleration. Let's define this acceleration.
Acceleration is directed to the same direction as the speed change, therefore, acceleration is directed to the center of the circle. An important assumption: the angle is so small that the chord length AB is the same as the arc length:
on two proportional sides and the angle between them. Consequently:
- centripetal acceleration module.Basics of dynamics Newton's first law. Inertial frames of reference. Galileo's principle of relativity
Any body remains motionless until other bodies act on it. A body moving at a certain speed continues to move evenly and rectilinearly until other bodies act on it. The Italian scientist Galileo Galilei first came to such conclusions about the laws of motion of bodies.
The phenomenon of maintaining the speed of body movement in the absence of external influences is called inertia.
All rest and movement of bodies are relative. One and the same body can be at rest in one frame of reference and move with acceleration in another. But there are such frames of reference, relative to which the translationally moving bodies keep their speed constant, if other bodies do not act on them... This statement is called Newton's first law (law of inertia).
The frames of reference relative to which the body, in the absence of external influences, moves rectilinearly and uniformly, are called inertial reference frames.
There can be any number of inertial reference frames, i.e. any frame of reference that moves uniformly and rectilinearly with respect to the inertial one is also inertial. There are no true (absolute) inertial reference systems.
The reason for the change in the speed of movement of bodies is always its interaction with other bodies.
When two bodies interact, the velocities of both the first and second bodies always change, i.e. both bodies acquire acceleration. The accelerations of two interacting bodies can be different, they depend on the inertia of the bodies.
Inertia- the body's ability to maintain its state of motion (rest). The more inertness of a body, the less acceleration it will acquire when interacting with other bodies, and the closer its motion will be to uniform rectilinear motion by inertia.
Weight- a physical quantity that characterizes the inertia of the body. The more mass a body has, the less acceleration it receives during interaction.
A kilogram is taken as a unit of mass in SI: [m] \u003d 1 kg.
In inertial reference frames, any change in the speed of a body occurs under the action of other bodies. ForceIs a quantitative expression of the action of one body on another.
Force- vector physical quantity, for its direction is taken the direction of the body's acceleration, which is caused by this force. Strength always has a point of application.
In SI, a unit of force is a force that imparts an acceleration of 1 m / s 2 to a body weighing 1 kg. This unit is called Newton:
.Newton's second law
The force acting on the body is equal to the product of the body mass by the acceleration imparted by this force:
.Thus, the acceleration of a body is directly proportional to the force acting on the body and inversely proportional to its mass:
.Since the linear speed uniformly changes direction, the movement along the circumference 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 under the grinder move, repeating the direction of 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, we can derive the following relations
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 the 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 rotational 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 that cause this acceleration can 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 ceases to act, then the body will move in a straight line
Consider moving a point on a circle from A to B. The linear velocity is
Now let us go over to a stationary system connected to the earth. 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.
Allows us to exist on this planet. How can you understand what constitutes centripetal acceleration? The definition of this physical quantity is presented below.
Observations
The simplest example of the acceleration of a body moving in a circle can be observed by rotating a stone on a rope. You pull the rope, and the rope pulls the stone towards the center. At each moment in time, the rope imparts a certain amount of movement to the stone, and each time in a new direction. You can imagine the movement of the rope as a series of weak jerks. A dash - and the rope changes its direction, another dash - another change, and so on in a circle. If you suddenly release the rope, the jerking will stop, and with it the change in the direction of speed will stop. The stone will move in a tangent direction to the circle. The question arises: "With what acceleration will the body move at this moment?"
Centripetal acceleration formula
First of all, it should be noted that the movement of the body in a circle is complex. The stone participates in two types of movement at the same time: under the action of force, it moves to the center of rotation, and at the same time it moves away from this center tangentially to the circle. According to Newton's Second Law, the force holding a stone on a rope is directed towards the center of rotation along that rope. The acceleration vector will be directed there.
Let for some time t our stone, moving uniformly with speed V, gets from point A to point B. Suppose that at the moment when the body crossed point B, the centripetal force ceased to act on it. Then, within a period of time, it would have got to point K. It lies on a tangent line. If at the same moment of time only centripetal forces acted on the body, then during time t, moving with the same acceleration, it would be at the point O, which is located on a straight line representing the diameter of the circle. Both segments are vectors and obey the vector addition rule. As a result of the summation of these two movements for a time interval t, we obtain the resultant movement along the arc AB.
If the time interval t is taken negligibly small, then the arc AB will differ little from the chord AB. Thus, it is possible to replace the arc movement with a chord movement. In this case, the movement of the stone along the chord will obey the laws straight motion, that is, the distance traveled AB will be equal to the product of the stone's speed and the time of its movement. AB \u003d V x t.
Let us denote the desired centripetal acceleration by the letter a. Then the path traversed only under the action of centripetal acceleration can be calculated by the formula uniformly accelerated motion:
Distance AB is equal to the product of speed and time, that is, AB \u003d V x t,
AO - calculated earlier by the formula of uniformly accelerated motion for moving in a straight line: AO \u003d at 2/2.
Substituting these data into the formula and transforming them, we get a simple and elegant formula for centripetal acceleration:
In words, it can be expressed as follows: the centripetal acceleration of a body moving in a circle is equal to the quotient of dividing the linear velocity in a square by the radius of the circle along which the body rotates. In this case, the centripetal force will look like in the picture below.
Angular velocity
The angular velocity is equal to the quotient of the linear velocity divided by the radius of the circle. The converse is also true: V \u003d ωR, where ω is the angular velocity
If you plug this value into the formula, you can get an expression for the centrifugal acceleration for the angular velocity. It will look like this:
Acceleration without changing speed
And yet, why does a body with acceleration directed towards the center not move faster and move closer to the center of rotation? The answer lies in the very wording of acceleration. Evidence suggests that circular motion is real, but requires acceleration toward the center to sustain it. Under the action of the force caused by this acceleration, the momentum changes, as a result of which the trajectory of movement is constantly curved, all the time changing the direction of the velocity vector, but not changing its absolute value. Moving in a circle, our long-suffering stone rushes inward, otherwise it would continue to move along a tangent. Every moment of time, leaving tangentially, the stone is attracted to the center, but does not fall into it. Another example of centripetal acceleration would be a water skier making small circles on the water. The figure of the athlete is tilted; he seems to fall, continuing to move and leaning forward.
Thus, we can conclude that acceleration does not increase the body's velocity, since the vectors of velocity and acceleration are perpendicular to each other. Adding to the velocity vector, acceleration only changes the direction of motion and keeps the body in orbit.
Exceeding the safety margin
In the previous experiment, we dealt with a perfect rope that did not break. But, let's say, our rope is the most common, and you can even calculate the effort, after which it will simply break. In order to calculate this force, it is enough to compare the margin of safety of the rope with the load that it experiences during the rotation of the stone. By rotating the stone at a faster speed, you give it more movement, and therefore more acceleration.
With a jute rope diameter of about 20 mm, its tensile strength is about 26 kN. It is noteworthy that the length of the rope does not appear anywhere. Rotating a weight of 1 kg on a rope with a radius of 1 m, we can calculate that the linear speed required to break it is 26 x 10 3 \u003d 1 kg x V 2/1 m.Thus, the speed that is dangerous to exceed will be equal to √ 26 x 10 3 \u003d 161 m / s.
Gravity
When considering the experiment, we neglected the effect of gravity, since at such high speeds its effect is negligible. But you can see that when untwisting a long rope, the body follows a more complex trajectory and gradually approaches the ground.
Celestial bodies
If you transfer the laws of motion in a circle to space and apply them to the motion of celestial bodies, you can rediscover several long-familiar formulas. For example, the force with which a body is attracted to the Earth is known by the formula:
In our case, the factor g is the very centripetal acceleration that was derived from the previous formula. Only in this case, the role of a stone will be played by a celestial body attracted to the Earth, and the role of a rope will be the force of gravity. The g factor will be expressed in terms of the radius of our planet and the speed of its rotation.
Outcomes
The essence of centripetal acceleration is the hard and thankless work of keeping a moving body in orbit. A paradoxical case is observed when the body does not change its velocity at constant acceleration. For an untrained mind, such a statement is rather paradoxical. Nevertheless, both when calculating the motion of an electron around the nucleus, and when calculating the speed of rotation of a star around a black hole, centripetal acceleration plays an important role.
In the study of motion in physics, the concept of a trajectory plays an important role. It is she who largely determines the type of movement of objects and, as a consequence, the type of formulas with which this movement is described. One of the common trajectories of movement is a circle. In this article, we will consider centripetal when moving in a circle.
Understanding full acceleration
Before characterizing centripetal acceleration when moving around a circle, let us consider the concept of full acceleration. It is believed physical quantity, which simultaneously describes the change in the value of the absolute and the velocity vector. In mathematical terms, this definition looks like this:
Acceleration is the full time derivative of speed.
As is known, the velocity v¯ of the body at each point of the trajectory is directed tangentially. This fact allows us to represent it in the form of the product of the modulus v and the unit tangent vector u¯, that is:
Then it can be calculated as follows:
a¯ \u003d d (v * u¯) / dt \u003d dv / dt * u¯ + v * du¯ / dt
The quantity a¯ is the vector sum of two terms. The first term is tangential (like the speed of a body) and is called It determines the rate of change in the modulus of speed. The second term - Let's consider it in more detail later in the article.
The expression obtained above for the normal component of acceleration a n ¯ can be written explicitly:
a n ¯ \u003d v * du¯ / dt \u003d v * du¯ / dl * dl / dt \u003d v 2 / r * r e ¯
Here dl is the path traversed by the body along the trajectory in time dt, r e ¯ is the unit vector directed to the center of curvature of the trajectory, r is the radius of this curvature. The resulting formula leads to several important features of the component a n¯ of the total acceleration:
- The quantity a n ¯ grows as the square of the velocity and decreases in inverse proportion to the radius, which distinguishes it from the tangential component. The latter is not equal to zero only if the speed module changes.
- Normal acceleration is always directed towards the center of curvature, which is why it is called centripetal.
Thus, the main condition for the existence of a nonzero quantity a n ¯ is the curvature of the trajectory. If such curvature does not exist (rectilinear displacement), then a n ¯ \u003d 0, since r-\u003e ∞.
Acceleration centripetal when moving in a circle
A circle is a geometric line, all points of which are at the same distance from some point. The latter is called the center of the circle, and the mentioned distance is its radius. If the speed of the body during rotation does not change in absolute value, then one speaks of an equally variable motion along a circle. In this case, the centripetal acceleration can be easily calculated using one of the two formulas below:
Where ω is the angular velocity, measured in radians per second (rad / s). The second equality is obtained thanks to the relationship between the angular and linear velocities:
Forces centripetal and centrifugal
When the body moves uniformly around the circumference, centripetal acceleration occurs due to the action of the corresponding centripetal force. Its vector is always directed towards the center of the circle.
The nature of this force can be very diverse. For example, when a person untwists a stone tied to a rope, the rope pulling force holds it on its trajectory. Another example of the action of centripetal force is the gravitational interaction between the Sun and the planets. It is it that makes all planets and asteroids move in circular orbits. The centripetal force is not able to change the kinetic energy of the body, since it is directed perpendicularly to its speed.
Each person could pay attention to the fact that when the car turns, for example, to the left, the passengers are pressed against the right edge of the vehicle interior. This process is the result of the centrifugal force of the rotational movement. In fact, this force is not real, since it is due to the inertial properties of the body and its tendency to move along a straight path.
Centrifugal and centripetal forces are equal in magnitude and opposite in direction. If this were not the case, then the circular trajectory of the body would be violated. If we take into account Newton's second law, then it can be argued that during rotational motion, centrifugal acceleration is equal to centripetal acceleration.