Speed and acceleration during curved motion. Movement of a body along a curved path
Let us consider the movement of a body along an arbitrary curvilinear trajectory. We have already noted above that when a body moves along a curvilinear trajectory, its velocity vector at any point is directed tangentially to the trajectory. The figure shows why this is so. The average speed is . This means that the direction of the average speed vector always coincides with the direction of movement Δr. But if we bring the end point closer to the start point, making the time interval Δt smaller and smaller, then, as can be seen from the figure, the direction of the vector Δr will approach the direction of the tangent to the trajectory at the initial point and in the limit will merge with it. But in this limit, the average speed will turn into instantaneous speed.
Unlike speed, acceleration when a body moves along a curved path is almost never directed tangentially to the path. Since , the direction of the acceleration vector always coincides with the direction of the velocity change vector. As can be seen from the figure, the vector of change in speed, and, therefore, acceleration is directed inside the curvature of the trajectory. In general, the angle between the velocity and acceleration vectors can vary from 0 to 180°.
Very often, the acceleration of a body when moving along a curved path is decomposed into two mutually perpendicular components: the direction tangent to the trajectory and the direction perpendicular to the tangent. The component of the total acceleration vector in the direction of the tangent to the trajectory is called tangential or tangential acceleration ( and τ). The component of the total acceleration vector in the direction perpendicular to the tangent is called centripetal or normal acceleration ( a c).
If α is the angle between the directions of acceleration and velocity, then we can write:
Besides:
The division of acceleration into two components is due to the fact that each component of the total acceleration characterizes the change in speed according to one of two parameters. Tangential acceleration characterizes the change in speed in magnitude. Tangential acceleration coincides in direction with the velocity vector if the velocity increases in magnitude and is directed opposite to the velocity if it decreases. When moving at a constant speed, the tangential acceleration is zero. The tangential acceleration module is equal to:
Centripetal acceleration characterizes the change in speed in direction. When moving along a straight path, the centripetal acceleration is zero.
An important special case of motion along a curved path is motion in a circle. The fact is that any smooth curved line can be replaced by a set of conjugate circular arcs of different radii. Let there be some curved line. At each point on a curve, many circles can be drawn that touch it at that point. But among all these circles there is one that better than others describes the curvature of the curve at a given point. The radius of this circle is called the radius of curvature of the line at that point. Thus, the movement of a body along an arbitrary curvilinear trajectory can be represented as sequential movement along circles of different radii.
Let the body move along a curved path. Let's consider two very close points of the trajectory A and B. Since the points are very close to each other, we can assume that they lie on a circular arc with a radius equal to the radius of curvature of the trajectory in this part of the trajectory - R. Let us assume that the speed of the body is constant in magnitude . In this case, the tangential acceleration is zero and the total acceleration of the body is equal to the centripetal acceleration. Triangle built on vectors v A, vB And Δv isosceles and similar to triangle AOB. So you can write:
Let Δt be the time during which the body moved from point A to point B. Since points A and B are located very close to each other (in the figure, for clarity, they are located far from each other), the chord AB practically coincides with the arc AB. Therefore you can write: . So we get:
Since tangential acceleration is zero, it represents centripetal acceleration. Thus, we obtain the formula for centripetal acceleration when a body moves along a curved path:
Here v is the instantaneous speed of the body, and R is the radius of curvature of the trajectory at a given point.
Curvilinear movements– movements whose trajectories are not straight, but curved lines. Planets and river waters move along curvilinear trajectories.
Curvilinear motion is always motion with acceleration, even if the absolute value of the velocity is constant. Curvilinear motion with constant acceleration always occurs in the plane in which the acceleration vectors and initial velocities of the point are located. In the case of curvilinear motion with constant acceleration in the plane xOy projections v x And v y its speed on the axis Ox And Oy and coordinates x And y points at any time t determined by formulas
A special case of curvilinear motion is circular motion. Circular motion, even uniform, is always accelerated motion: the velocity module is always directed tangentially to the trajectory, constantly changing direction, so circular motion always occurs with centripetal acceleration where r– radius of the circle.
The acceleration vector when moving in a circle is directed towards the center of the circle and perpendicular to the velocity vector.
In curvilinear motion, acceleration can be represented as the sum of normal and tangential components:
Normal (centripetal) acceleration is directed towards the center of curvature of the trajectory and characterizes the change in speed in the direction:
v – instantaneous speed value, r– radius of curvature of the trajectory at a given point.
Tangential (tangential) acceleration is directed tangentially to the trajectory and characterizes the change in speed modulo.
The total acceleration with which a material point moves is equal to:
In addition to centripetal acceleration, the most important characteristics of uniform circular motion are the period and frequency of revolution.
Circulation period- this is the time during which the body completes one revolution .
The period is indicated by the letter T(c) and is determined by the formula:
Where t- circulation time, n- the number of revolutions completed during this time.
Frequency- this is a quantity numerically equal to the number of revolutions completed per unit of time.
Frequency is denoted by a Greek letter (nu) and is found using the formula:
The frequency is measured in 1/s.
Period and frequency are mutually inverse quantities:
If a body moves in a circle with speed v, makes one revolution, then the distance traveled by this body can be found by multiplying the speed v for the time of one revolution:
l = vT. On the other hand, this path is equal to the circumference of the circle 2π r. That's why
vT = 2π r,
Where w(s -1) - angular velocity.
At a constant rotation frequency, centripetal acceleration is directly proportional to the distance from the moving particle to the center of rotation.
Angular velocity (w) – a value equal to the ratio of the angle of rotation of the radius at which the rotating point is located to the period of time during which this rotation occurred:
.
Relationship between linear and angular speeds:
The movement of a body can be considered known only when it is known how each of its points moves. The simplest motion of solid bodies is translational. Progressive is the motion of a rigid body in which any straight line drawn in this body moves parallel to itself.
Kinematics - it's easy!
A description of the movement of a body is considered complete when it is known how each point moves.
In general, any complex motion of a rigid (undeformed) body can be represented as the sum of two motions: translational and rotational. Forward movement- if any straight line drawn inside the body moves parallel to itself.
During the translational motion of a rigid body, all its points have the same speeds, accelerations, displacements and trajectories.
The forward movement can also be curvilinear.
To describe the translational motion of a body, it is enough to create an equation of motion for one of its points, then the calculations are simplified.
In curvilinear motion, the body moves along a curved path.
In general curvilinear trajectory is a collection of sections of circular arcs of different diameters.
In curvilinear motion, the velocity and acceleration vectors not directed along one straight line.
A special case of curvilinear motion is uniform motion in a circle.
Uniform movement of a point around a circle
Circular motion is the simplest type of curvilinear motion.
When a point moves uniformly around a circle:
The speed of movement V in a circle is called linear speed,
A moving point passes a circle of equal length in equal intervals of time.
The velocity vector at any point of the trajectory is directed tangentially to her.
At each point of the trajectory, the acceleration vector is directed radially towards the center of the circle.
This acceleration is called centripetal acceleration.
The module of centripetal acceleration is equal to:
Where
a c - centripetal acceleration, [m/s2];
υ - linear speed, [m/s];
R - radius of the circle, [m].
The path traveled by a point moving uniformly around a circle for any period of time t is equal to:
For one full revolution around the circle, i.e. in a time equal to the period T, the point travels a path equal to the circumference
In this case, the linear speed of the point is equal to:
The velocity vector and the centripetal acceleration vector are always mutually perpendicular.
Velocity and acceleration remain constant in absolute value, but change their direction.
The uniform movement of a point around a circle is motion with variable acceleration, since the acceleration continuously changes in direction.
Rotational motion of a rigid body around a fixed axis
During rotational motion around a fixed axis, all points of the body describe circles centered on the axis of rotation of the body.
Each point has its own speed, acceleration and displacement.
Characteristics of rotational motion
1. Angular velocity- this is the ratio of the angle of rotation to the time during which it occurs.
The letter designation for angular velocity is omega.
where are the units of measurement
If a body moves uniformly, then any point of this body rotates through the same angle over the same period of time.
2. Rotational speed is the number of revolutions per unit time.
3. Rotation period- this is the time of one full revolution.
4. When rotating, a full revolution is
Then
5. Linear speed is the speed of a point moving in a circle.
Each point of a rotating body has its own linear speed.
Uniformly accelerated curvilinear motion
Curvilinear movements are movements whose trajectories are not straight, but curved lines. Planets and river waters move along curvilinear trajectories.
Curvilinear motion is always motion with acceleration, even if the absolute value of the velocity is constant. Curvilinear motion with constant acceleration always occurs in the plane in which the acceleration vectors and initial velocities of the point are located. In the case of curvilinear motion with constant acceleration in the xOy plane, the projections vx and vy of its velocity on the Ox and Oy axes and the x and y coordinates of the point at any time t are determined by the formulas
Uneven movement. Rough speed
No body moves at a constant speed all the time. When the car starts moving, it moves faster and faster. It can move steadily for a while, but then it slows down and stops. In this case, the car travels different distances in the same time.
Movement in which a body travels unequal lengths of path in equal intervals of time is called uneven. With such movement, the speed does not remain unchanged. In this case, we can only talk about average speed.
Average speed shows how much movement a body goes through per unit time. It is equal to the ratio of the displacement of the body to the time of movement. Average speed, like the speed of a body during uniform motion, is measured in meters divided by a second. In order to characterize motion more accurately, instantaneous speed is used in physics.
The speed of a body at a given moment in time or at a given point in the trajectory is called instantaneous speed. Instantaneous speed is a vector quantity and is directed in the same way as the displacement vector. You can measure instantaneous speed using a speedometer. In the International System, instantaneous speed is measured in meters divided by second.
point movement speed uneven
Movement of a body in a circle
Curvilinear motion is very common in nature and technology. It is more complex than a straight line, since there are many curved trajectories; this movement is always accelerated, even when the velocity module does not change.
But movement along any curved path can be approximately represented as movement along the arcs of a circle.
When a body moves in a circle, the direction of the velocity vector changes from point to point. Therefore, when they talk about the speed of such movement, they mean instantaneous speed. The velocity vector is directed tangentially to the circle, and the displacement vector is directed along the chords.
Uniform circular motion is a motion during which the module of the motion velocity does not change, only its direction changes. The acceleration of such motion is always directed towards the center of the circle and is called centripetal. In order to find the acceleration of a body moving in a circle, it is necessary to divide the square of the speed by the radius of the circle.
In addition to acceleration, the circular motion of a body is characterized by the following quantities:
The period of rotation of a body is the time during which the body makes one complete revolution. The rotation period is designated by the letter T and is measured in seconds.
The frequency of rotation of a body is the number of revolutions per unit time. Is the rotation speed indicated by a letter? and is measured in hertz. In order to find the frequency, you need to divide one by the period.
Linear speed is the ratio of the movement of a body to time. In order to find the linear speed of a body in a circle, it is necessary to divide the circumference by the period (the circumference is equal to 2? multiplied by the radius).
Angular velocity is a physical quantity equal to the ratio of the angle of rotation of the radius of the circle along which the body moves to the time of movement. Angular velocity is indicated by a letter? and is measured in radians divided per second. Can you find the angular velocity by dividing 2? for a period. Angular velocity and linear velocity among themselves. In order to find the linear speed, it is necessary to multiply the angular speed by the radius of the circle.
Figure 6. Circular motion, formulas.
Depending on the shape of the trajectory, movement is divided into rectilinear and curvilinear. In the real world, we most often deal with curvilinear motion, when the trajectory is a curved line. Examples of such movement are the trajectory of a body thrown at an angle to the horizon, the movement of the Earth around the Sun, the movement of the planets, the end of a clock hand on a dial, etc.
Figure 1. Trajectory and displacement during curved motion
Definition
Curvilinear motion is a motion whose trajectory is a curved line (for example, a circle, ellipse, hyperbola, parabola). When moving along a curvilinear trajectory, the displacement vector $\overrightarrow(s)$ is directed along the chord (Fig. 1), and l is the length of the trajectory. The instantaneous speed of the body (that is, the speed of the body at a given point of the trajectory) is directed tangentially at the point of the trajectory where the moving body is currently located (Fig. 2).
Figure 2. Instantaneous speed during curved motion
However, the following approach is more convenient. This movement can be represented as a combination of several movements along circular arcs (see Fig. 4.). There will be fewer such partitions than in the previous case; in addition, the movement along the circle is itself curvilinear.
Figure 4. Breakdown of curvilinear motion into motion along circular arcs
Conclusion
In order to describe curvilinear movement, you need to learn to describe movement in a circle, and then represent arbitrary movement in the form of sets of movements along circular arcs.
The task of studying the curvilinear motion of a material point is to compile a kinematic equation that describes this motion and allows, based on given initial conditions, to determine all the characteristics of this motion.