Everyone paid attention to the whole variety of types of movement that he encounters in his life. However, any mechanical movement of the body is reduced to one of two types: linear or rotational. We 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 straight path or with a circular one. Any other trajectories can be obtained thanks to a combination of these two. Further in the article, the following laws of body motion will be considered:
- Uniform in a straight line.
- Equally accelerated (equidistant) in a straight line.
- Uniform in circumference.
- Equally accelerated in a circle.
- Movement along an elliptical trajectory.
Uniform movement, or rest
From a scientific point of view, Galileo became interested in this movement for the first time at the end of the 16th - beginning of the 17th centuries. Studying the inertial properties of the body, as well as introducing the concept of a reference frame, he guessed that the state of rest and uniform movement are one and the same thing (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 the body, according to which the speed of the latter is a constant value whenever there are no external forces that change the characteristics of motion.
Uniform rectilinear movement of the body in space is described by the following formula:
s = v * t
Where s is the distance that the body will cover in time t, moving at a speed v. This simple expression is also written in the following forms (it all depends on the quantities that are known):
v = s / t; t = s / v
Linear acceleration
According to Newton’s second law, the presence of an external force acting on the body inevitably leads to the appearance of acceleration in the latter. From the definition of acceleration (speed of change of speed) follows the expression:
a = v / t or v = a * t
If the external force acting on the body remains constant (will not change the modulus and direction), then the acceleration will not change either. This type of motion is called uniformly accelerated, where acceleration acts as a proportionality coefficient between speed and time (speed increases linearly).
For this movement, the distance traveled is calculated by integrating speed over time. The law of motion of the body for the path with uniformly accelerated movement takes the form:
s = a * t 2/2
The most common example of this movement is the fall of any object from a height at which gravity gives it an acceleration of g = 9.81 m / s 2 .
Rectilinear accelerated (slowed down) movement with the presence of initial speed
In fact, we are talking about a combination of the two types of movement discussed in the previous paragraphs. Imagine a simple situation: the car was traveling at a certain speed v 0 , then the driver pressed the brakes, and after a while the vehicle stopped. How to describe the movement in this case? For the function of speed versus time, the expression
v = v 0 - a * t
Here v 0 is the initial speed (before the car decelerates). 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), we obtain the formula for the path:
s = v 0 * t - a * t 2/2
Note that according to this formula, only the braking distance is calculated. To find out the distance traveled by the car for the entire duration of its movement, you should find the sum of two paths: for uniform and equally slow motion.
In the example described above, if the driver did not press the brake pedal, but the gas pedal, then in the formulas presented the sign “-” would change to “+”.
Circular motion
Any movement in a circle cannot occur without acceleration, because even when the speed 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 = v 2 / r, r is the radius
In this expression, the speed may depend on time, as is the case in the case of uniformly accelerated circular motion. 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 make significant efforts to unwind a projectile before throwing it.
Rotation around the axis at a constant speed
This type of movement is identical to the previous one, only it is customary to describe it using linear physical quantities, but using angular characteristics. The law of rotational motion of the body, when the angular velocity does not change, is written in scalar form as follows:
L = I * ω
Here L and I are the angular momentum and inertia moments, respectively, ω is the angular velocity, which is connected with the linear equation:
v = ω * r
The value of ω shows how many radians the body rotates in a second. The quantities L and I have the same meaning as momentum and mass for rectilinear motion. Accordingly, the angle θ at which the body rotates in time t is calculated as follows:
θ = ω * t
An example of this type of movement is the rotation of a flywheel located on a crankshaft in a car engine. A flywheel is a massive disk, which is very difficult to give any acceleration. Thanks to this, it provides a smooth change in the torque that is transmitted from the engine to the wheels.
Rotation around the axis with acceleration
If an external force is applied to a system that is able to rotate, then it will begin to increase its angular velocity. This situation is described by the following law of motion of the body about the axis of rotation :
F * d = I * dω / dt
Here F is an external force that is applied to the system at a distance d from the axis of rotation. The work on the left side of the equality is called the moment of power.
For uniformly accelerated circular motion, we obtain that ω depends on time as follows:
ω = α * t, where α = F * d / I is the angular acceleration
In this case, the rotation angle in time t can be determined by integrating ω over time, that is:
θ = α * t 2/2
If the body already rotated at 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 = ω 0 + α * t;
θ = ω 0 * t + α * t 2/2
Thus, the appearance of an external moment of forces is the cause of the presence of acceleration in a system with an axis of rotation.
For completeness of information, we note that the rotation speed ω can be changed not only using the 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 the skaters on ice. Grouping, athletes increase ω by decreasing I, according to the simple law of body movement:
I * ω = const
Elliptical trajectory on the example of the planets of the solar system
As you know, our Earth and other planets of the solar system do not revolve around their stars in a circle, but in 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 planetary motion, Kepler came to 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 in one of the focal points of the ellipse.
- The radius vector that connects the Sun and the planet, for equal periods of time, describes the same area. This fact follows from the conservation of angular momentum.
- If we divide the square of the period of revolution by the cube of the semimajor axis of the elliptical orbit of the planet, we get some constant that is the same for all the planets of our system. Mathematically, it is written like this:
T 2 / a 3 = C = const
Subsequently, Isaac Newton, using these laws of motion of bodies (planets), formulated his famous law of universal gravity, or gravity. Using it, it can be shown that the constant C in Kepler’s 3rd law is equal to:
C = 4 * pi 2 / (G * M)
Where G is the gravitational universal constant, and M is the mass of the Sun.
Note that the motion in an elliptical orbit in the case of the action of a central force (gravity) leads to the fact that the linear velocity v is constantly changing. It is maximum when the planet is closest to the star, and minimal away from it.