The movement of the body under the action of gravity is one of the central topics in dynamic physics. The fact that the dynamics section is based on the three laws of Newton, even an ordinary student knows. Let's try to analyze this topic thoroughly, and an article that describes each example in detail will help us make the study of body movement under the influence of gravity as useful as possible.
A bit of history
From time immemorial, people have watched with curiosity the various phenomena occurring in our lives. Mankind for a long time could not understand the principles and structure of many systems, but a long way to study the world around us led our ancestors to a scientific revolution. Nowadays, when technologies are developing at an incredible speed, people almost do not think about how certain mechanisms work.
Meanwhile, our ancestors were always interested in the mysteries of natural processes and the structure of the world, searched for answers to the most complex questions, and did not stop studying until they found answers to them. So, for example, the famous scientist Galileo Galilei in the 16th century asked himself the questions: "Why do bodies always fall down, what kind of force pulls them to the ground?" In 1589, he set up a series of experiments, the results of which were very valuable. He studied in detail the patterns of free fall of various bodies, dropping objects from the famous tower in the city of Pisa. The laws that he derived were improved and described in more detail by the formulas of another well-known English scientist - Sir Isaac Newton. It is he who owns the three laws on which almost all modern physics is based.
The fact that the laws of motion of bodies described more than 500 years ago are still relevant today means that our planet is subject to unchanging laws. Modern man needs to at least superficially study the basic principles of the arrangement of the world.
Basic and auxiliary concepts of dynamics
In order to fully understand the principles of such a movement, you should first familiarize yourself with some concepts. So, the most necessary theoretical terms:
- Interaction is the effect of bodies on each other, in which there is a change or the beginning of their movement relative to each other. There are four types of interaction: electromagnetic, weak, strong and gravitational.
- Speed is a physical quantity denoting the speed with which a body moves. Speed is a vector, that is, it has not only significance, but also direction.
- Acceleration is the value that shows us the speed of a change in the speed of a body over a period of time. It is also a vector quantity.
- The path is a curve, and sometimes a straight line, which the body outlines when moving. With uniform rectilinear motion, the trajectory may coincide with the displacement value.
- The path is the length of the trajectory, that is, exactly as much as the body has passed in a certain amount of time.
- An inertial reference system is a medium in which Newton’s first law is fulfilled, that is, the body retains its inertia, provided that all external forces are completely absent.
The above concepts are quite enough to correctly draw or imagine in the head a simulation of the movement of the body under the influence of gravity.
What does power mean?
Let's move on to the basic concept of our topic. So, force is a quantity, the meaning of which is the impact or influence of one body on another quantitatively. And gravity is that force that acts absolutely on every body located on the surface or near our planet. The question arises: where does this very force come from? The answer lies in the law of gravity.
And what is gravity?
Any body from the side of the Earth is influenced by a gravitational force, which tells it some acceleration. Gravity always has a vertical direction down to the center of the planet. In other words, gravity pulls objects to the Earth, which is why objects always fall down. It turns out that gravity is a special case of gravity. Newton derived one of the main formulas for finding the force of attraction between two bodies. It looks like this: F = G * (m 1 x m 2 ) / R 2 .
What is the acceleration of gravity?
A body that is released from a certain height always flies down under the influence of gravity. The motion of the body under the action of gravity vertically up and down can be described by equations where the acceleration value "g" will be the main constant. This value is solely due to the action of the attractive force, and its value is approximately equal to 9.8 m / s 2 . It turns out that a body thrown from a height without an initial speed will move down with acceleration equal to the value of "g".
The movement of the body under the action of gravity: formulas for solving problems
The basic formula for finding gravity is as follows: F gravity = m x g, where m is the mass of the body affected by the force, and "g" is the acceleration of gravity (to simplify the tasks, it is assumed to be 10 m / s 2 ) .
There are several more formulas used to find this or that unknown with the free movement of the body. So, for example, in order to calculate the path traveled by the body, it is necessary to substitute the known values in this formula: S = V 0 x t + a x t 2/2 (the path is equal to the sum of the products of the initial velocity times the time and the acceleration by the square of time divided by 2).
Equations for describing the vertical movement of the body
The vertical motion of the body under the action of gravity can be described by the equation, which looks like this: x = x 0 + v 0 x t + a x t 2 / 2. Using this expression, we can find the coordinates of the body at a known point in time. You just need to substitute the values known in the problem: initial location, initial speed (if the body was not just released, but pushed with some force) and acceleration, in our case it will be equal to the acceleration g.
In the same way, you can find the speed of a body that moves under the action of gravity. The expression for finding an unknown quantity at any moment of time: v = v 0 + g x t (the value of the initial velocity can be zero, then the velocity will be equal to the product of the acceleration of gravity by the time in which the body makes the movement).
The movement of bodies under the action of gravity: problems and methods for solving them
When solving many problems related to gravity, we recommend using the following plan:
- To determine a convenient inertial reference frame for yourself, it is usually customary to choose the Earth, because it meets many of the requirements for ISO.
- Draw a small drawing or drawing that depicts the main forces acting on the body. The motion of a body under the action of gravity implies a sketch or diagram, which indicates in which direction the body moves if acceleration equal to g acts on it.
- Then you should choose the direction for projecting the forces and the obtained accelerations.
- Record unknown quantities and determine their direction.
- And finally, using the above formulas to solve problems, calculate all unknown quantities by substituting data in the equations for finding the acceleration or the distance traveled.
Ready-made solution to an easy task
When it comes to such a thing as the movement of a body under the action of gravity, determining which way is more practical to solve the problem can be difficult. However, there are several tricks, using which you can easily solve even the most difficult task. So, let’s take a look at living examples of how to solve a particular problem. Let's start with an easy-to-understand task.
Some body was released from a height of 20 m without initial velocity. Determine how much time it will reach the surface of the earth.
Solution: we know the path traveled by the body, we know that the initial speed was 0. We can also determine that only gravity acts on the body, it turns out that this is the movement of the body under the influence of gravity, and therefore we should use this formula: S = V 0 x t + a x t 2/2. Since in our case a = g, after some transformations we get the following equation: S = g x t 2 / 2. Now it remains only to express the time in terms of this formula, we obtain that t 2 = 2S / g. We substitute the known values (we assume that g = 10 m / s 2 ) t 2 = 2 x 20/10 = 4. Therefore, t = 2 s.
So our answer is: the body will fall to the ground in 2 seconds.
The trick that allows you to quickly solve the problem is as follows: you can notice that the described body movement in the given problem occurs in one direction (vertically down). It is very similar to uniformly accelerated motion, since no force other than gravity acts on the body (we neglect the force of air resistance). Thanks to this, you can use the easy formula to find the path with uniformly accelerated movement, bypassing the image of the drawings with the alignment of forces acting on the body.
An example of solving a more complex problem
And now let's see how it is best to solve the problems of body motion under the influence of gravity, if the body does not move vertically, but has a more complex character of movement.
For example, the following task. An object of mass m moves with unknown acceleration down an inclined plane whose friction coefficient is k. Determine the value of acceleration, which is available during the movement of a given body, if the inclination angle α is known.
Solution: You should use the plan described above. First of all, draw a drawing of the inclined plane with the image of the body and all the forces acting on it. It turns out that three components act on it: gravity, friction, and support reaction force. The general equation of the resultant forces looks like this: F friction + N + mg = ma.
The main highlight of the problem is the condition of inclination at an angle α. When projecting forces on the ox axis and the oy axis, it is necessary to take this condition into account, then we get the following expression: mg x sin α - F friction = ma (for the axis ox) and N - mg x cos α = F friction (for the axis oy) .
Friction F is easy to calculate by the formula for finding the friction force, it is equal to k x mg (coefficient of friction, multiplied by the product of body mass and acceleration of gravity). After all the calculations, it remains only to substitute the found values into the formula, we get a simplified equation for calculating the acceleration with which the body moves along an inclined plane.