The normal acceleration vector of a moving point is directed. Acceleration. Normal and tangential acceleration. Plane-parallel motion of a rigid body

Let's consider some of the simplest types of point movement, often encountered in practice.

Uniform motion of a point is its motion with a constant algebraic speed or

where C is the integration constant.

Let at the initial moment of time the position of point M on the trajectory be then characterized by

Thus, with uniform motion, the path traversed by a point depends linearly on time.

Uniform motion of a point

Equally variable motion of a point is such a motion of it in which the algebraic value of the tangential acceleration remains constant:

If the sign of a coincides with the sign of the speed, then the motion is called uniformly accelerated. If the signs of a and do not coincide, the motion is called equally slow. From the last equality we have:

where is the constant of integration. If then

Thus, with uniform motion, speed depends linearly on time. Rewriting the last equality as:

where is the constant of integration. Determining from the condition that when we find

Thus, with uniform motion, the path traversed by a point is a square trinomial in t.

Circular motion of a point

The movement of a point in a circle or circular motion is often encountered in practice. Let point M move along a circle of radius R counterclockwise (Fig. 24). Counting the arc from some initial position of the point, we write it through the central angle in the form:

The algebraic speed of the point will be:

where - is called the angular velocity of the point and is denoted by ω, its dimension is .

Using the concept of angular velocity, we write:

Hence, the speed of a point in a circular motion is equal to the product of the radius of the trajectory and the angular speed.

The tangential acceleration of a point is:

where - is called angular acceleration and is denoted by its dimension,

The normal acceleration of the point will be:

Since it is directed towards the center of the circle, it is often called centripetal. The modulus of the total acceleration of a point is equal to

When a point moves uniformly around a circle, therefore, there is no tangential acceleration in this case and there is only a constant centripetal acceleration.

With uniform circular motion

Physical meaning of tangential and normal acceleration of a point

The introduction of the concept of uniform and uniformly variable motion of a point allows us to indicate the physical meaning of the tangential and normal acceleration of a point. Indeed, let the tangential acceleration be zero everywhere:

Then, if then from the last equality we have:

or the point moves at a constant speed, i.e. the point moves uniformly.

From this we can conclude that the presence of tangential acceleration characterizes the uneven movement of a point along the trajectory. Let further normal acceleration be equal to zero:

Then, if that normal acceleration can be identically equal to zero only in the case when

or the trajectory of a point is a straight line - the movement is rectilinear.

Thus, the absence of normal acceleration for a certain period of time indicates straightness of movement. From this we can conclude that the presence of normal acceleration indicates the curvature of the trajectory.

If at the same time the tangential and normal accelerations are identically zero, then the motion of the point will be uniform and rectilinear. If only at a single moment in time the tangential acceleration is zero, then this indicates that on the graph of the function the extrema of the function or its inflection points correspond to this moment. If only at a single moment in time the normal acceleration is zero, then this indicates that at that moment the speed of the moving point is zero or the radius of curvature of the trajectory is equal to infinity.

Point acceleration for all 3 ways to accelerate movement

The acceleration of a point characterizes the speed of change in the magnitude and direction of the point's velocity.

1. Acceleration of a point when specifying its movement in a vector way

the acceleration vector of a point is equal to the first derivative of the velocity or the second derivative of the radius vector of the point with respect to time. The acceleration vector is directed towards the concavity of the curve

2. Acceleration of a point when specifying its movement using the coordinate method

The magnitude and direction of the acceleration vector are determined from the relations:

3. Determination of acceleration when specifying its movement in a natural way

Natural axes and natural trihedron

Natural axes. Curvature characterizes the degree of curvature (curvature) of a curve. Thus, a circle has a constant curvature, which is measured by the value K, the reciprocal of the radius,

The larger the radius, the smaller the curvature, and vice versa. A straight line can be considered as a circle with an infinitely large radius and a curvature of zero. The point represents a circle of radius R = 0 and has infinitely large curvature.

An arbitrary curve has variable curvature. At each point of such a curve, you can select a circle with a radius whose curvature is equal to the curvature of the curve at a given point M (Fig. 9.2). The quantity is called the radius of curvature at a given point on the curve. The axis directed tangentially in the direction of movement and the axis directed radially to the center of curvature and called the normal form the natural coordinate axes.

Normal and tangential acceleration of a point

In the natural way of defining motion, the acceleration of a point is equal to the geometric sum of two vectors, one of which is directed along the main normal and is called normal acceleration, and the second is directed along a tangent and is called the tangential acceleration of the point.

The projection of the acceleration of a point onto the main normal is equal to the square of the modulus of the velocity of boredom divided by the radius of curvature of the trajectory at the corresponding point. The normal acceleration of a point is always directed towards the center of curvature of the trajectory and is equal in magnitude to this projection.

The change in speed modulo is characterized by tangential (tangential) acceleration.

those. the projection of the acceleration of a point onto the tangent is equal to the second derivative of the arc coordinate of the point with respect to time or the first derivative of the algebraic value of the speed of the point with respect to time.

This projection has a plus sign if the directions of the tangential acceleration and the unit vector coincide, and a minus sign if they are opposite.

Thus, in the case of a natural method of specifying movement, when the trajectory of a point and, consequently, its radius of curvature are known? at any point and the equation of motion, you can find the projections of the point’s acceleration onto the natural axes:

If a > 0 and > 0 or a< 0 и < 0, то движение ускоренное и вектор а направлен в сторону вектора скорости. Если а < 0 и >0 or a > 0 and< 0, то движение замедленное и вектор а направлен в сторону, противоположную вектору скорости

Special cases.

1. If a point moves rectilinearly and unevenly, then = , and, consequently, = 0, a = a.

2. If a point moves rectilinearly and uniformly, = 0, a = 0 and a = 0.

3. If a point moves uniformly along a curved path, then a = 0 and a = . With uniform curvilinear motion of a point, the law of motion has the form s = t. It is advisable to assign a positive reference direction in tasks depending on specific conditions. In the case when 0 = 0, we get = gt and. Often in problems the formula is used (when a body falls from a height H without an initial speed)

Conclusion: normal acceleration exists only at curvilinear

32. Classification of the movement of a point by its acceleration

if during a certain period of time the normal and tangential accelerations of a point are equal to zero, then during this interval neither the direction nor the magnitude of the velocity will change, i.e. the point moves uniformly in a straight line and its acceleration is zero.

if for a certain period of time the normal acceleration is not zero and the tangential acceleration of a point is zero, then the direction of the velocity changes without changing its module, i.e. the point moves curvilinearly uniformly and the acceleration module.

If at a single moment in time, then the point does not move uniformly, and at this moment in time the modulus of its speed has a maximum, minimum, or the smallest rate of monotonic change.

if for a certain period of time the normal acceleration of a point is zero and the tangent acceleration is not zero, then the direction of the velocity does not change, but its magnitude changes, i.e. the point moves unevenly in a straight line. Point acceleration module in this case

Moreover, if the directions of the velocity vectors coincide, then the motion of the point is accelerated, and if they do not coincide, then the motion of the point is slow.

If at some point in time, then the point does not move rectilinearly, but passes the inflection point of the trajectory or the modulus of its velocity becomes zero.

If for a certain period of time neither the normal nor the tangential acceleration are equal to zero, then both the direction and the magnitude of its velocity change, i.e. the point makes a curvilinear uneven movement. Point acceleration module

Moreover, if the directions of the velocity vectors coincide, then the movement is accelerated, and if they are opposite, then the movement is slow.

If the tangential acceleration module is constant, i.e. , then the modulus of the point’s velocity changes proportionally to time, i.e. the point undergoes uniform motion. And then

Formula for the speed of uniformly variable motion of a point;

Equation of uniform motion of a point

In order to be able to solve various problems on the movement of bodies in physics, you need to know the definitions of physical quantities, as well as the formulas by which they are related. This article will cover the questions of what tangential velocity is, what total acceleration is, and what components make it up.

Concept of speed

The two main quantities of the kinematics of movement of bodies in space are speed and acceleration. Speed describes the speed of movement, so the mathematical form for it is as follows:

The concept of acceleration

In contrast to speed, which characterizes the speed at which a body passes a trajectory, acceleration is a quantity that describes the rate of change in speed, which is written mathematically as follows:

Like speed, acceleration is a vector characteristic. However, its direction is not related to the velocity vector. It is determined by the change in direction v¯. If during the movement the speed does not change its vector, then the acceleration a¯ will be directed along the same line as the speed. This acceleration is called tangential. If the speed changes direction while maintaining its absolute value, then the acceleration will be directed towards the center of curvature of the trajectory. It's called normal.

Acceleration is measured in m/s2. For example, the well-known acceleration of gravity is tangential when an object rises or falls vertically. Its value near the surface of our planet is 9.81 m/s2, that is, for every second of fall, the speed of the body increases by 9.81 m/s.

The cause of acceleration is not speed, but force. If a force F acts on a body of mass m, then it will inevitably create an acceleration a, which can be calculated as follows:

This formula is a direct consequence of Newton's second law.

Full, normal and tangential acceleration

Speed and acceleration as physical quantities were discussed in previous paragraphs. Now we will study in more detail which components make up the total acceleration a¯.

Let us assume that the body moves with speed v¯ along a curved path. Then the equality will be true:

The vector u¯ has unit length and is directed along the tangent line to the trajectory. Using this representation of the velocity v¯, we obtain the equality for the total acceleration:

a¯ = dv¯/dt = d(v*u¯)/dt = dv/dt*u¯ + v*du¯/dt.

The first term obtained in the right equality is called tangential acceleration. Velocity is related to it by the fact that it quantifies the change in the absolute value of v¯ without taking into account its direction.

The second term is the normal acceleration. It quantitatively describes the change in the velocity vector, without taking into account the change in its modulus.

If we denote the tangential and normal components of the total acceleration a as at and an, then the modulus of the latter can be calculated using the formula:

a = √(at2 + an2).

Relationship between tangential acceleration and speed

The corresponding connection is described by kinematic expressions. For example, in the case of motion in a straight line with constant acceleration, which is tangential (the normal component is zero), the following expressions are valid:

In the case of circular motion with constant acceleration, these formulas are also valid.

Thus, whatever the trajectory of the body’s movement, the tangential acceleration through the tangential velocity is calculated as the time derivative of its module, that is:

For example, if the speed changes according to the law v = 3*t3 + 4*t, then at will be equal to:

at = dv/dt = 9*t2 + 4.

Speed and normal acceleration

We write explicitly the formula for the normal component an, we have:

an¯ = v*du¯/dt = v*du¯/dl*dl/dt = v2/r*re¯

Where re¯ is a vector of unit length, which is directed towards the center of curvature of the trajectory. This expression establishes the relationship between tangential velocity and normal acceleration. We see that the latter depends on the modulus v at a given time and on the radius of curvature r.

Normal acceleration appears whenever the velocity vector changes, but it is zero if this vector remains in the same direction. It makes sense to talk about the quantity an¯ only when the curvature of the trajectory is a finite quantity.

We noted above that when moving in a straight line, there is no normal acceleration. However, in nature there is a type of trajectory along which an has a finite value, and at = 0 for |v¯| = const. This trajectory is a circle. For example, rotation at a constant frequency of a metal shaft, carousel or planet around its own axis occurs with constant normal acceleration an and zero tangential acceleration at.

And why is it needed? We already know what a reference system, relativity of motion and a material point are. Well, it's time to move on! Here we will look at the basic concepts of kinematics, put together the most useful formulas for the basics of kinematics, and give a practical example of solving the problem.

Let's solve this problem: a point moves in a circle with a radius of 4 meters. The law of its motion is expressed by the equation S=A+Bt^2. A=8m, B=-2m/s^2. At what point in time is the normal acceleration of a point equal to 9 m/s^2? Find the speed, tangential and total acceleration of the point for this moment in time.

Solution: we know that in order to find the speed we need to take the first time derivative of the law of motion, and the normal acceleration is equal to the quotient of the square of the speed and the radius of the circle along which the point is moving. Armed with this knowledge, we will find the required quantities.

Need help solving problems? Professional student service is ready to provide it.

The study of physics begins with the consideration of mechanical motion. In the general case, bodies move along curved trajectories with variable speeds. The concept of acceleration is used to describe them. In this article we will look at what tangential and normal acceleration are.

Kinematic quantities. Speed and acceleration in physics

Kinematics of mechanical motion is a branch of physics that deals with the study and description of the movement of bodies in space. Kinematics operates on three main quantities:

distance traveled;
speed;
acceleration.

In the case of motion in a circle, similar kinematic characteristics are used, which are reduced to the central angle of the circle.

Everyone is familiar with the concept of speed. It shows the speed of change in the coordinates of bodies in motion. The speed is always directed tangentially to the line along which the body moves (trajectory). In what follows, we will denote the linear velocity by v¯, and the angular velocity by ω¯.

Acceleration is the rate of change of the quantities v¯ and ω¯. Acceleration is also, but its direction is completely independent of the velocity vector. Acceleration is always directed towards the force acting on the body, which causes a change in the velocity vector. Acceleration for any type of movement can be calculated using the formula:

The more the speed changes over the time interval dt, the greater the acceleration will be.

Tangential and normal acceleration

Suppose that a material point moves along some curved line. It is known that at some moment t its speed was equal to v¯. Since velocity is a vector tangent to the trajectory, it can be represented in the following form:

Here v is the length of the vector v¯, and u t ¯ is the unit velocity vector.

To calculate the total acceleration vector at time t, it is necessary to find the time derivative of the velocity. We have:

a¯ = dv¯ / dt = d (v × u t ¯) / dt

Since the velocity module and the unit vector change with time, using the rule for finding the derivative of the product of functions, we obtain:

a¯ = dv / dt × u t ¯ + d (u t ¯) / dt × v

The first term in the formula is called the tangential, or tangential component of acceleration, the second term is normal acceleration.

Tangential acceleration

Let's write the formula for calculating tangential acceleration again:

a t ¯ = dv / dt × u t ¯

This equality means that the tangential (tangential) acceleration is directed in the same way as the velocity vector at any point of the trajectory. It numerically determines the change in velocity modulus. For example, in the case of rectilinear motion it consists only of a tangential component. Normal acceleration for this type of movement is zero.

The reason for the appearance of the value a t ¯ is the influence of an external force on a moving body.

In the case of rotation with constant angular acceleration α, the tangential component of acceleration can be calculated using the following formula:

Here r is the radius of rotation of the material point under consideration, for which the value a t is calculated.

Normal or centripetal acceleration

Now let’s write out the second component of the total acceleration again:

a c ¯ = d (u t ¯) / dt × v

From geometric considerations, it can be shown that the time derivative of a unit tangent to the trajectory of a vector is equal to the ratio of the velocity modulus v to the radius r at time t. Then the expression above will be written like this:

This formula for normal acceleration indicates that, unlike the tangential component, it does not depend on changes in speed, but is determined by the square of the modulus of the speed itself. Also, a c increases with decreasing radius of rotation at a constant value of v.

Normal acceleration is called centripetal because it is directed from the center of mass of a rotating body to the axis of rotation.

The reason for the appearance of this acceleration is the central component of the force acting on the body. For example, in the case of planets revolving around our Sun, the centripetal force is gravitational attraction.

Normal acceleration of a body changes only the direction of speed. It is not capable of changing its module. This fact is an important difference from the tangential component of the total acceleration.

Since centripetal acceleration always occurs when the velocity vector rotates, it also exists in the case of uniform circular rotation, in which the tangential acceleration is zero.

In practice, you can feel the effects of normal acceleration if you are in the car when it makes a long turn. In this case, passengers are pressed against the direction of rotation of the car door. This phenomenon is the result of the action of two forces: centrifugal (displacement of passengers from their seats) and centripetal (pressure on passengers from the side of the car door).

Module and direction of total acceleration

So, we have found out that the tangential component of the physical quantity under consideration is directed tangentially to the trajectory of motion. In turn, the normal component is perpendicular to the trajectory at a given point. This means that the two acceleration components are perpendicular to each other. Their vector addition gives the total acceleration vector. Its module can be calculated using the following formula:

a = √(a t 2 + a c 2)

The direction of the vector a¯ can be determined both relative to the vector a t ¯ and relative to a c ¯. To do this, use the appropriate trigonometric function. For example, the angle between full and normal acceleration is:

Solving the problem of determining centripetal acceleration

A wheel, which has a radius of 20 cm, spins with an angular acceleration of 5 rad/s 2 for 10 seconds. It is necessary to determine the normal acceleration of points located on the periphery of the wheel after a specified time.