Everyone paid attention to all the variety of types of movement that he encounters in his life. However, any mechanical movement of a body is reduced to one of two types: linear or rotational. Consider in the article the basic laws of motion of bodies.
What types of movement will be discussed?
As noted in the introduction, all types of body motion that are considered in classical physics are associated either with a rectilinear trajectory or with a circular one. Any other trajectory can be obtained by combining the two. Further in the article the following laws of body motion will be considered:
- Uniform in a straight line.
- Equally accelerated (equally slowed down) in a straight line.
- Uniform around the circumference.
- Equally accelerated around the circumference.
- Movement along an elliptical path.
Uniform movement, or a state of rest
From a scientific point of view, Galileo first became interested in this movement in the late 16th - early 17th centuries. Studying the inertial properties of a body, as well as introducing the concept of a frame of reference, he guessed that the state of rest and uniform motion is one and the same (it all depends on the choice of the object relative to which the speed is calculated).
Subsequently, Isaac Newton formulated his first law of motion of a body, according to which the speed of the latter is always constant when there are no external forces that change the characteristics of motion.
Uniform rectilinear movement of a body in space is described by the following formula:
Where s is the distance that the body will cover in time t, moving with speed v. This simple expression is also written in the following forms (it all depends on the quantities that are known):
Travel in a straight line with acceleration
According to Newton's second law, the presence of an external force acting on a body inevitably leads to the appearance of acceleration in the latter. From (the rate of change of speed) the expression follows:
a \u003d v / t or v \u003d a * t
If the external force acting on the body remains constant (will not change the modulus and direction), then the acceleration will also not change. This type of motion is called uniformly accelerated, where acceleration is a coefficient of proportionality between speed and time (the speed increases linearly).
For this movement, the distance traveled is calculated by integrating the speed over time. The law of motion of a body for a path with uniformly accelerated movement takes the form:
The most common example of this movement is the fall of any object from a height, in which the force of gravity gives it an acceleration g \u003d 9.81 m / s 2.
Rectilinear accelerated (decelerated) motion with an initial speed
In fact, we are talking about a combination of the two types of movement discussed in the previous paragraphs. Imagine simple situation: the car was driving at a certain speed v 0, then the driver applied the brakes, and the vehicle stopped after a while. How to describe the movement in this case? For the function of speed versus time, the following expression is valid:
Here v 0 is the initial speed (before braking the car). The minus sign indicates that the external force (sliding friction) is directed against the speed v 0.
As in the previous paragraph, if we take the time integral of v (t), then we obtain the formula for the path:
s \u003d v 0 * t - a * t 2/2
Note that this formula only calculates the braking distance. To find out the distance traveled by the car during its entire movement, you should find the sum of two paths: for uniform and for equally slow motion.
In the example described above, if the driver pressed not on the brake pedal, but on the gas pedal, then in the presented formulas the sign "-" would change to "+".
Circular motion
Any movement along a circle cannot occur without acceleration, since even if the velocity modulus is preserved, its direction changes. The acceleration associated with this change is called centripetal (it is it that bends the trajectory of the body, turning it into a circle). The module of this acceleration is calculated as follows:
a c \u003d v 2 / r, r - radius
In this expression, the speed can depend on time, as it happens in the case of equals accelerated movement around the circumference. In the latter case, a c will grow rapidly (quadratic dependence).
Centripetal acceleration determines the force that must be applied to keep the body in a circular orbit. An example is the hammer throw competition, where athletes put significant effort into spinning the apparatus before throwing it.
Rotation around an axis at a constant speed
This type of movement is identical to the previous one, only it is customary to describe it not using linear physical quantities, and using angular characteristics. The law of rotational motion of a body, when the angular velocity does not change, is written in scalar form as follows:
Here L and I are the moments of impulse and inertia, respectively, ω is the angular velocity, which is related to the linear one by the equality:
The value of ω shows how many radians the body will rotate in a second. The quantities L and I have the same meaning as momentum and mass for straight motion... Accordingly, the angle θ through which the body will rotate in time t is calculated as follows:
An example of this type of motion is the rotation of a flywheel located on the crankshaft in a car engine. A flywheel is a massive disc that is very difficult to give any acceleration. Thanks to this, it provides a smooth change in torque, which is transmitted from the engine to the wheels.
Rotation around an axis with acceleration
If an external force is applied to a system that is capable of rotating, then it will begin to increase its angular velocity. This situation is described by the following law of body motion around:
Here F is the external force applied to the system at a distance d from the axis of rotation. The product on the left side of the equality is called the moment of force.
For uniformly accelerated motion along a circle, we find that ω depends on time as follows:
ω \u003d α * t, where α \u003d F * d / I - angular acceleration
In this case, the angle of rotation in time t can be determined by integrating ω over time, that is:
If the body was already rotating with a certain speed ω 0, and then the external moment of force F * d began to act, then by analogy with the linear case one can write the following expressions:
ω \u003d ω 0 + α * t;
θ \u003d ω 0 * t + α * t 2/2
Thus, the appearance of an external torque is the reason for the presence of acceleration in a system with an axis of rotation.
For completeness of information, we note that it is possible to change the rotation speed ω not only with the help of an external moment of forces, but also due to a change in the internal characteristics of the system, in particular, its moment of inertia. This situation was seen by every person who watched the rotation of skaters on the ice. Grouping, athletes increase ω due to a decrease in I, according to a simple law of body movement:
Movement along an elliptical trajectory on the example of the planets of the solar system
As you know, our Earth and other planets Solar system revolve around their star not in a circle, but along an elliptical trajectory. For the first time, mathematical laws to describe this rotation were formulated by the famous German scientist Johannes Kepler at the beginning of the 17th century. Using the results of his teacher Tycho Brahe's observations of the motion of the planets, Kepler came up with the formulation of his three laws. They are formulated as follows:
- The planets of the solar system move in elliptical orbits, with the sun located at one of the focuses of the ellipse.
- The radius vector, which connects the Sun and the planet, describes the same areas at equal intervals of time. This fact follows from the conservation of angular momentum.
- If we divide the square of the orbital period by the cube of the semi-major axis of the planet's elliptical orbit, then we get some constant, which is the same for all planets of our system. Mathematically, it is written like this:
T 2 / a 3 \u003d С \u003d const
Subsequently, Isaac Newton, using these laws of motion of bodies (planets), formulated his famous law of universal gravity, or gravitation. Applying it, one can show that the constant C in the 3rd is equal to:
C \u003d 4 * pi 2 / (G * M)
Where G is the gravitational universal constant and M is the mass of the Sun.
Note that motion in an elliptical orbit in the case of the action of a central force (gravitation) leads to the fact that the linear velocity v is constantly changing. It is maximal when the planet is closest to the star, and minimal when far from it.
And why is it needed. We already know what a frame of reference is, a relativity of motion and a material point. Well, it's time to move on! Here we will look at the basic concepts of kinematics, bring together the most useful formulas on the basics of kinematics and give a practical example of solving the problem.
Let's solve the following problem: the point moves in a circle with a radius of 4 meters. The law of its motion is expressed by the equation S \u003d A + Bt ^ 2. A \u003d 8m, B \u003d -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 a 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 values.
Need help solving problems? Professional student service is ready to provide it.
THE DERIVATIVE AND ITS APPLICATION TO THE STUDY OF FUNCTIONS X
Section 218. Law of traffic. Instant movement speed
A more complete description of the movement can be obtained as follows. We divide the time of body movement into several separate intervals ( t 1 , t 2), (t 2 , t 3), etc. (not necessarily equal, see Fig. 309) and on each of them we set the average speed of movement.
These average speeds, of course, will more fully characterize the movement throughout the entire section than average speed for the entire time of movement. However, they will not give an answer to such, for example, a question: at what point in time in the interval from t 1 to t 2 (Fig. 309) the train went faster: at the moment t " 1 or at the moment t " 2 ?
The more fully the average speed characterizes the movement, the shorter the sections of the path on which it is determined. Therefore, one of the possible ways to describe the uneven movement is to set the average speeds of this movement on increasingly small sections of the path.
Suppose the function s (t ), indicating which path the body travels, moving rectilinearly in the same direction, during the time t from the beginning of the movement. This function determines the law of body motion. For example, uniform motion occurs according to the law
s (t ) = vt ,
where v - movement speed; free fall of bodies occurs according to the law
where g - the acceleration of a freely falling body, etc.
Consider the path traveled by a body moving according to a certain law s (t ), for the time from t before t + τ .
By the time t body go the way s (t ), and by the time t + τ - way s (t + τ ). Therefore, for the time from t before t + τ it will travel a path equal to s (t + τ ) - s (t ).
Dividing this path by the time of movement τ , we get the average speed for the time from t before t + τ :
The limit of this speed at τ -\u003e 0 (if only it exists) is called instantaneous speed of movement at time t:
(1)
The instantaneous speed of movement at the moment of time t is called the limit of the average speed of movement from t before t+ τ when τ tends to zero.
Let's look at two examples.
Example 1... Uniform movement in a straight line.
In this case s (t ) = vt where v - movement speed. Let's find the instantaneous speed of this movement. To do this, you first need to know the average speed in the time interval from t before t + τ ... But for uniform movement, the average speed on any part of the turbidity coincides with the speed of movement v ... Therefore instantaneous speed v (t ) will be equal to:
v (t ) =v = v
So, for a uniform movement, the instantaneous speed (as well as the average speed on any part of the path) coincides with the speed of movement.
Of course, the same result could be reached formally, proceeding from equality (1).
Really,
Example 2. Equally accelerated motion with zero initial speed and acceleration and ... In this case, as is known from physics, the body moves according to the law
By formula (1), we obtain that the instantaneous speed of such movement v (t ) is equal to:
So, the instantaneous speed of uniformly accelerated motion at the moment of time t equal to acceleration times time t ... In contrast to uniform motion, the instantaneous speed of uniformly accelerated motion changes over time.
Exercises
1741. The Point Moves According to the Law 
(s
- path in meters, t
- time in minutes). Find the instantaneous speed of this point:
b) at time t 0 .
1742. Find the instantaneous speed of a point moving according to the law s (t ) = t 3 (s - path in meters, t - time in minutes):
a) at the initial moment of movement;
b) 10 seconds after the start of the movement;
c) at the moment t \u003d 5 minutes;
1743. Find the instantaneous speed of a body moving according to the law s (t ) = √t , at an arbitrary moment of time t .
Let's consider one more particular problem.
It is known that the body's velocity module during the entire movement remained constant and equal to 5 m / s. Find the law of motion of this body. The origin of the path lengths coincides with the starting point of body movement.
To solve the problem, we use the formula
From here, you can find the increment of the path length for any small time interval
By condition, the speed module is constant. This means that the increments in the path length for any equal time intervals will be the same. By definition, this is a uniform movement. The equation we have obtained is nothing more than the law of such uniform motion. If expressions are substituted into this equation, then it is easy to obtain
Let's assume that the origin of time coincides with the beginning of body movement. Let us take into account that, according to the condition, the origin of the path lengths coincides with the starting point of the body movement. We take as an interval the time from the beginning of the motion to the moment we need.Then we must put After substitution of these values, the law of the considered motion will have the form
The example considered allows us to give a new definition of uniform motion (§ 13): uniform motion is motion with a constant modulus of velocity.
The same example allows you to obtain a general formula for the law of uniform motion.
If the origin of time coincides with the beginning of movement, and the origin of the path lengths coincides with the initial point of movement, then the law of uniform movement will have the form
If the start time of movement and the length of the path to the initial point of movement, then the law of uniform movement takes on a more complex form:
Let us pay attention to one more important result that can be obtained from the law of uniform motion we have found. Suppose that for some uniform movement a graph of the dependence of speed on time is given (Fig. 1.60). The law of this motion It can be seen from the figure that the product is numerically equal to the area of \u200b\u200bthe figure bounded by the coordinate axes, the graph of the dependence of the speed on time and the ordinate corresponding
at a given moment in time, according to the speed graph, you can calculate the increments of path lengths while driving.
Using a more complex mathematical apparatus, it can be shown that this result, obtained by us for a particular case, turns out to be valid for any uneven movements. The increment of the path length during the movement is always numerically equal to the area of \u200b\u200bthe figure, limited by the velocity graph by the coordinate axes and the ordinate corresponding to the selected end point in time.
This possibility of graphically finding the law of complex movements will be used in the future.