Centripetal acceleration is the acceleration component of a point characterizing the change in the direction of the velocity vector for a trajectory with curvature. (The second component, tangential acceleration, is characterized by a change in the modulus of velocity.) It is directed towards the center of curvature of the trajectory, 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 generally equivalent to the term " normal acceleration"; the differences are only stylistic (sometimes historical).
The simplest example of centripetal acceleration is the acceleration vector for uniform circular motion (directed towards the center of the circle).
Elementary formula
where is the normal (centripetal) acceleration, is the (instantaneous) linear speed of movement along the trajectory, is the (instantaneous) angular velocity of this movement relative to the center of curvature of the trajectory, is the radius of curvature of the trajectory at a given point. (The connection between the first formula and the second is obvious given).
The expressions above include absolute values. They can be easily written in vector form by multiplying by - the unit vector from the center of curvature of the trajectory to its given point:
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 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), 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. This is aggravated by the fact that when moving with a constant value of speed, the tangential component will be 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 motion along a circle (which, moreover, can be generalized to the general case with practically no change).
Geometric derivation for irregular circular motion
Geometric inference for free motion (along free path)
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 through the unit tangent vector:
By the 19th century, consideration of centripetal acceleration had become completely routine for both pure science and engineering applications.
Since the linear speed uniformly changes direction, the movement around the circle cannot be called uniform, it is uniformly 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 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 the movement of a point on a circle from A to B. The linear velocity is equal to v A and v B respectively. Acceleration - the change in speed per unit of time. Let's find the difference in vectors.
Centripetal acceleration is the acceleration component of a point, which characterizes 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". The 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 circular motion (directed towards the center of the circle).
Blast Acceleration in projection onto a plane perpendicular to the axis, it appears as centripetal.
Collegiate YouTube
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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:
an = 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 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 = d v / d t (\ displaystyle a _ (\ tau) = 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 = v e τ (\ displaystyle \ mathbf (v) = v \, \ mathbf (e) _ (\ tau)) through the unit tangent vector e τ (\ displaystyle \ mathbf (e) _ (\ tau)):
a = dvdt = d (ve τ) dt = dvdte τ + vde τ dt = dvdte τ + vde τ dldldt = dvdte τ + v 2 R en, (\ 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 used the notation for the unit normal vector to the trajectory and l (\ displaystyle l \)- for the current length of the trajectory ( l = l (t) (\ displaystyle l = l (t) \)); the last transition also used the obvious
d l / d t = v (\ displaystyle dl / dt = v \)and, for geometric reasons,
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 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 - with an accuracy of the second order of smallness in terms of distance to a given point).
Story
Huygens was apparently the first to obtain the correct formulas for centripetal acceleration (or centrifugal force). Practically from this time on, 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 the dependence of the gravitational force on the distance to the source of gravity, based on Kepler's third law derived from observations).
By the 19th century, consideration of centripetal acceleration had become completely routine for both pure science and engineering applications.