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.
Most simple example centripetal acceleration is the acceleration vector during uniform circular motion (directed towards the center of the circle).
Rapid acceleration in projection onto a plane perpendicular to the axis, it appears as centripetal.
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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:
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 (according to 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 vector instantaneous speed); the full acceleration vector then also includes a tangential component ( tangential acceleration) a τ = d v / d t (\displaystyle a_(\tau )=dv/dt\ ), the 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 trajectory of the vector ( tangential acceleration) and another orthogonal to it (normal acceleration) - can be convenient and useful, 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\ ).
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 for 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 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 from it - will coincide with the given curve - trajectory - up to the second order of smallness in the distance to the given point).
Story
Apparently, Huygens was the first to obtain the correct formulas for centripetal acceleration (or centrifugal force). 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).
TO 19th century consideration of centripetal acceleration is already becoming completely routine both for pure science and for engineering applications.
For example, a car that starts moving moves faster as it increases its speed. At the point where the motion begins, the speed of the car is zero. Having started moving, the car accelerates to a certain speed. If you need to brake, the car will not be able to stop instantly, but rather over time. That is, the speed of the car will tend to zero - the car will begin to move slowly until it stops completely. But physics does not have the term “slowdown”. If a body moves, decreasing speed, this process is also called acceleration, but with a “-” sign.
Medium acceleration is called the ratio of the change in speed to the period of time during which this change occurred. Calculate the average acceleration using the formula:
where is this . The direction of the acceleration vector is the same as the direction of change in speed Δ = - 0
where 0 is the initial speed. At a moment in time t 1(see figure below) at the body 0. At a moment in time t 2 the body has speed. Based on the rule of vector subtraction, we determine the vector of speed change Δ = - 0. From here we calculate the acceleration:
.
In the SI system unit of acceleration called 1 meter per second per second (or meter per second squared):
.A meter per second squared is the acceleration of a rectilinearly moving point, at which the speed of this point increases by 1 m/s in 1 second. In other words, acceleration determines the degree of change in the speed of a body in 1 s. For example, if the acceleration is 5 m/s2, then the speed of the body increases by 5 m/s every second.
Instantaneous acceleration of a body (material point) V at the moment time is a physical quantity that is equal to the limit to which the average acceleration tends as the time interval tends to 0. In other words, this is the acceleration developed by the body in a very short period of time:
.Acceleration has the same direction as the change in speed Δ in extremely short periods of time during which the speed changes. The acceleration vector can be specified using projections onto the corresponding coordinate axes in a given reference system (projections a X, a Y, a Z).
With accelerated linear motion, the speed of the body increases in absolute value, i.e. v 2 > v 1 , and the acceleration vector has the same direction as the velocity vector 2 .
If the speed of a body decreases in absolute value (v 2< v 1), значит, у вектора ускорения направление противоположно направлению вектора скорости 2 . Другими словами, в таком случае наблюдаем slowing down(acceleration is negative, and< 0). На рисунке ниже изображено направление векторов ускорения при прямолинейном движении тела для случая ускорения и замедления.
If movement occurs along a curved path, then the magnitude and direction of the speed changes. This means that the acceleration vector is depicted as two components.
Tangential (tangential) acceleration they call that component of the acceleration vector that is directed tangentially to the trajectory at a given point of the motion trajectory. Tangential acceleration describes the degree of change in speed modulo during curvilinear motion.
U tangential acceleration vectorτ (see figure above) the direction is the same as that of linear speed or opposite to it. Those. the tangential acceleration vector is in the same axis with the tangent circle, which is the trajectory of the body.
Acceleration in kinematics formula. Acceleration in kinematics definition.
What is acceleration?
Speed may change while driving.
Velocity is a vector quantity.
The velocity vector can change in direction and magnitude, i.e. in size. To account for such changes in speed, acceleration is used.
Acceleration definition
Definition of acceleration
Acceleration is a measure of any change in speed.
Acceleration, also called total acceleration, is a vector.
Acceleration vector
The acceleration vector is the sum of two other vectors. One of these other vectors is called tangential acceleration, and the other is called normal acceleration.
Describes the change in the magnitude of the velocity vector.
Describes the change in direction of the velocity vector.
When moving in a straight line, the direction of speed does not change. In this case, the normal acceleration is zero, and the total and tangential accelerations coincide.
With uniform motion, the velocity module does not change. In this case, the tangential acceleration is zero, and the total and normal accelerations are the same.
If a body performs rectilinear uniform motion, then its acceleration is zero. And this means that the components of total acceleration, i.e. normal acceleration and tangential acceleration are also zero.
Full acceleration vector
The total acceleration vector is equal to the geometric sum of the normal and tangential accelerations, as shown in the figure:
Acceleration formula:
a = a n + a t
Full acceleration module
Full acceleration module:
Angle alpha between the total acceleration vector and normal acceleration (aka the angle between the total acceleration vector and the radius vector):
Please note that the total acceleration vector is not directed tangentially to the trajectory.
The tangential acceleration vector is directed along the tangent.
The direction of the total acceleration vector is determined by the vector sum of the normal and tangential acceleration vectors.
Normal distribution is the most common type of distribution. One encounters it when analyzing measurement errors, monitoring technological processes and modes, as well as in the analysis and prediction of various phenomena in biology, medicine and other fields of knowledge.
The term “normal distribution” is used in a conditional sense as generally accepted in the literature, although not entirely successful. Thus, the statement that some attribute obeys normal law distribution does not at all mean the presence of any unshakable norms that supposedly underlie the phenomenon of which the attribute in question is a reflection, and subordination to other laws of distribution does not mean some kind of abnormality of this phenomenon.
The main feature of the normal distribution is that it is the limit to which other distributions approach. The normal distribution was first discovered by Moivre in 1733. Only continuous random variables obey the normal law. The density of the normal distribution law has the form .
The mathematical expectation for the normal distribution law is . The variance is equal to .
Basic properties of normal distribution.
1. The distribution density function is defined on the entire numerical axis Oh , that is, each value X corresponds to a very specific value of the function.
2. For all values X (both positive and negative) the density function takes positive values, that is, the normal curve is located above the axis Oh .
3. Limit of the density function with unlimited increase X equal to zero,
.4. The normal distribution density function at a point has a maximum
.5. The graph of the density function is symmetrical about the straight line.
6. The distribution curve has two inflection points with coordinates
And 
.7. The mode and median of the normal distribution coincide with the mathematical expectation A .
8. The shape of the normal curve does not change when changing the parameter A .
9. The coefficients of skewness and kurtosis of the normal distribution are equal to zero.
The importance of calculating these coefficients for empirical distribution series is obvious, since they characterize the skewness and steepness of this series in comparison with the normal one.
The probability of falling into the interval is found by the formula
, Where 
odd tabulated function.Let us determine the probability that a normally distributed random variable deviates from its mathematical expectation by an amount less than , that is, we will find the probability of the inequality
, or the probability of double inequality. Substituting into the formula, we get
Expressing the deviation of a random variable X in fractions of the standard deviation, that is, putting in the last equality, we get .
Then when we get ,
when we get ,
when we receive .
From the last inequality it follows that practically the scattering of a normally distributed random variable is confined to the area . The probability that a random variable will not fall into this area is very small, namely equal to 0.0027, that is, this event can only occur in three cases out of 1000. Such events can be considered almost impossible. Based on the above reasoning rule of three sigma, which is formulated as follows: if a random variable has a normal distribution, then the deviation of this value from the mathematical expectation in absolute value does not exceed three times the standard deviation.
Example 28. A part produced by an automatic machine is considered suitable if the deviation of its controlled size from the design one does not exceed 10 mm. Random deviations of the controlled size from the design are subject to the normal distribution law with a standard deviation of mm and mathematical expectation. What percentage of suitable parts does the machine produce?
Solution. Consider the random variable X - deviation of the size from the design one. The part will be considered valid if the random variable belongs to the interval. The probability of producing a suitable part can be found using the formula
. Consequently, the percentage of suitable parts produced by the machine is 95.44%.Binomial distribution
Binomial is the probability distribution of occurrence m number of events in n independent trials, in each of which the probability of an event occurring is constant and equal to r . The probability of the possible number of occurrences of an event is calculated using the Bernoulli formula: ,
Where . Permanent n And r , included in this expression, are the parameters of the binomial law. The binomial distribution describes the probability distribution of a discrete random variable.
Basic numerical characteristics binomial distribution. The mathematical expectation is . The variance is
. The coefficients of skewness and kurtosis are equal and 
. With an unlimited increase in the number of tests A
And E
tend to zero, therefore, we can assume that the binomial distribution converges to normal as the number of trials increases.Example 29. Produced independent tests with the same probability of occurrence of the event A in every test. Find the probability of an event occurring A in one trial if the variance of the number of occurrences across three trials is 0.63.
Solution. For binomial distribution
. Let's substitute the values, we get 
from here 
or 
then and .Poisson distribution
Law of distribution of rare phenomena
The Poisson distribution describes the number of events m , occurring over equal periods of time, provided that events occur independently of each other with a constant average intensity. Moreover, the number of tests n is high, and the probability of the event occurring in each trial r small Therefore, the Poisson distribution is called the law of rare events or the simplest flow. The Poisson distribution parameter is the value characterizing the intensity of occurrence of events in n tests. Poisson distribution formula
.The Poisson distribution well describes the number of claims for payment of insurance amounts per year, the number of calls received at the telephone exchange in a certain time, the number of failures of elements during reliability tests, the number of defective products, and so on.
Basic numerical characteristics for the Poisson distribution. The mathematical expectation is equal to the variance and is equal to A . That is
. This is distinctive feature this distribution. The coefficients of skewness and kurtosis are respectively equal 
.Example 30. The average number of insurance payments per day is two. Find the probability that in five days you will have to pay: 1) 6 insurance amounts; 2) less than six amounts; 3) at least six.distribution.
This distribution is often observed when studying service life various devices, the time of failure-free operation of individual elements, parts of the system and the system as a whole, when considering random time intervals between the occurrence of two consecutive rare events.
The density of the exponential distribution is determined by the parameter, which is called failure rate. This term is associated with a specific application area - reliability theory.
The expression for the integral function of the exponential distribution can be found using the properties of the differential function:
Expectation of exponential distribution, variance, mean standard deviation. Thus, it is characteristic of this distribution that the standard deviation is numerically equal to the mathematical expectation. For any value of the parameter, the coefficients of asymmetry and kurtosis are constant values
.Example 31. The average operating time of a TV before the first failure is 500 hours. Find the probability that a randomly selected TV will operate without breakdowns for more than 1000 hours.
Solution. Since the average operating time to first failure is 500, then
. We find the desired probability using the formula.Linear movement, linear speed, linear acceleration.
Moving(in kinematics) - change of location physical body in space relative to the chosen reference system. The vector characterizing this change is also called displacement. It has the property of additivity. The length of the segment is the displacement module, measured in meters (SI).
You can define movement as a change in the radius vector of a point: .
The displacement module coincides with the distance traveled if and only if the direction of displacement does not change during movement. In this case, the trajectory will be a straight line segment. In any other case, for example, when curvilinear movement, from the triangle inequality it follows that the path is strictly longer.
Vector D r = r -r 0 drawn from the initial position of the moving point to its position at a given time (increment of the radius vector of the point over the considered period of time) is called moving.
During rectilinear motion, the displacement vector coincides with the corresponding section of the trajectory and the displacement module |D r| equal to the distance traveled D s.
Linear speed of a body in mechanicsSpeed
To characterize the motion of a material point, a vector quantity is introduced - speed, which is defined as rapidity movement and his direction at a given moment in time.
Let a material point move along some curvilinear trajectory so that at the moment of time t it corresponds to the radius vector r 0 (Fig. 3). For a short period of time D t the point will go along the path D s and will receive an elementary (infinitesimal) displacement Dr.
Average speed vector
is called the ratio of the increment Dr of the radius vector of a point to the time interval D t: The direction of the average velocity vector coincides with the direction of Dr. With an unlimited decrease in D t the average speed tends to a limiting value called instantaneous speed v:
Instantaneous speed v, therefore, is a vector quantity equal to the first derivative of the radius vector of the moving point with respect to time. Since the secant in the limit coincides with the tangent, the velocity vector v is directed tangent to the trajectory in the direction of motion (Fig. 3). As D decreases t path D s will increasingly approach |Dr|, so the absolute value of the instantaneous velocity
Thus, the absolute value of the instantaneous speed is equal to the first derivative of the path with respect to time:
At uneven movement - the module of instantaneous speed changes over time. In this case, we use the scalar quantity b vñ - average speed uneven movement:
From Fig. 3 it follows that á vñ> |ávñ|, since D s> |Dr|, and only in the case of rectilinear motion
If expression d s = v d t(see formula (2.2)) integrate over time ranging from t to t+D t, then we find the length of the path traveled by the point in time D t:
In case uniform motion numeric value instantaneous speed constantly; then expression (2.3) will take the form
The length of the path traveled by a point during the period of time from t 1 to t 2, given by the integral
Acceleration and its components
In the case of uneven movement, it is important to know how quickly the speed changes over time. Physical size, which characterizes the rate of change in speed in magnitude and direction, is acceleration.
Let's consider flat movement, those. a movement in which all parts of a point’s trajectory lie in the same plane. Let the vector v specify the speed of the point A at a point in time t. During time D t the moving point has moved to position IN and acquired a speed different from v both in magnitude and direction and equal to v 1 = v + Dv. Let's move the vector v 1 to the point A and find Dv (Fig. 4).
Medium acceleration uneven movement in the range from t to t+D t is a vector quantity equal to the ratio of the change in speed Dv to the time interval D t
Instant acceleration and (acceleration) of a material point at the moment of time t there will be a limit of average acceleration:
Thus, acceleration a is a vector quantity equal to the first derivative of speed with respect to time.
Let us decompose the vector Dv into two components. To do this from the point A(Fig. 4) in the direction of velocity v we plot the vector equal in absolute value to v 1 . Obviously, the vector , equal to , determines the change in speed over time D t modulo: . The second component of the vector Dv characterizes the change in speed over time D t in direction.
Tangential and normal acceleration.
Tangential acceleration- acceleration component directed tangentially to the motion trajectory. Coincides with the direction of the velocity vector at accelerated movement and in the opposite direction when slow. Characterizes the change in speed module. It is usually designated or (, etc. in accordance with which letter is chosen to denote acceleration in general in this text).
Sometimes tangential acceleration is understood as the projection of the tangential acceleration vector - as defined above - onto the unit vector of the tangent to the trajectory, which coincides with the projection of the (full) acceleration vector onto the unit tangent vector, that is, the corresponding expansion coefficient in the accompanying basis. In this case, not a vector notation is used, but a “scalar” one - as usual for the projection or coordinates of a vector - .
The magnitude of the tangential acceleration - in the sense of the projection of the acceleration vector onto a unit tangent vector of the trajectory - can be expressed as follows:
where is the ground speed along the trajectory, coinciding with the absolute value of the instantaneous speed at a given moment.
If we use the notation for the unit tangent vector, then we can write the tangential acceleration in vector form:
Conclusion
The expression for tangential acceleration can be found by differentiating with respect to time the velocity vector, represented in terms of the unit tangent vector:
where the first term is the tangential acceleration, and the second is the normal acceleration.
Here we use the notation for the unit vector of the normal to the trajectory and - for the current length of the trajectory (); the last transition also uses the obvious
and, from geometric considerations,
Centripetal acceleration(normal)- part of the total acceleration of a point, due to the curvature of the trajectory and the speed of movement of the material point along it. This acceleration is directed towards the center of curvature of the trajectory, which is what gives rise to the term. Formally and essentially, the term centripetal acceleration generally coincides with the term normal acceleration, differing rather only stylistically (sometimes historically).
Particularly often they talk about centripetal acceleration when we are talking about uniform motion in a circle or when moving more or less close to this particular case.
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 - a unit vector from the center of curvature of the trajectory to a 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 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), the direction coinciding with the tangent to the trajectory (or, what is the same, with the instantaneous speed).
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. This is aggravated by the fact that when moving at a constant speed, the tangential component will be equal to zero, that is, in this important particular case, only the normal component remains. 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 particular case of motion in a circle (which, moreover, can be generalized to the general case with virtually no changes).
