Two beams emanating from it form an angle. Its value can be specified in both radians and degrees. Now, at some distance from the center point, let's mentally draw a circle. The measure of the angle, expressed in radians, in this case is the mathematical ratio of the length of the arc L, separated by two rays, to the value of the distance between the center point and the circle line (R), that is:
If we now imagine the described system as material, then not only the concepts of angle and radius, but also centripetal acceleration, rotation, etc. can be applied to it. Most of them describe the behavior of a point on a rotating circle. By the way, a solid disk can also be represented by a set of circles, the difference of which is only in the distance from the center.
One of the characteristics of such a rotating system is the period of revolution. It indicates the time it takes for a point on an arbitrary circle to return to its original position or, which is also true, to rotate 360 degrees. At a constant rotation speed, the correspondence is T = (2 * 3.1416) / Ug (hereinafter, Ug is the angle).
The rotational speed indicates the number of complete revolutions performed in 1 second. At a constant speed, we get v = 1 / T.
Depends on time and the so-called angle of rotation. That is, if we take an arbitrary point A on the circle as the origin, then during the rotation of the system this point will shift to A1 in time t, forming an angle between the radii A-center and A1-center. Knowing the time and angle, you can calculate the angular velocity.
And since there is a circle, motion and speed, then there is also centripetal acceleration. It is one of the components describing the movement in the case of curvilinear motion. The terms "normal" and "centripetal acceleration" are identical. The difference is that the second one is used to describe movement in a circle when the acceleration vector is directed towards the center of the system. Therefore, it is always necessary to know exactly how the body (point) moves and its centripetal acceleration. Its definition is as follows: it is the rate of change of speed, the vector of which is directed perpendicular to the direction of the vector and changes the direction of the latter. The encyclopedia indicates that Huygens was engaged in the study of this issue. The formula for centripetal acceleration proposed by him looks like this:
Acs = (v*v) / r,
where r is the radius of curvature of the path traveled; v - movement speed.
The formula by which centripetal acceleration is calculated is still hotly debated among enthusiasts. For example, a curious theory was recently voiced.
Huygens, considering the system, proceeded from the fact that the body moves in a circle of radius R with a speed v measured at the starting point A. Since the inertia vector is directed along, a trajectory in the form of a straight line AB is obtained. However, the centripetal force keeps the body on a circle at point C. If we designate the center as O and draw the lines AB, BO (the sum of BS and CO), as well as AO, we get a triangle. According to the Pythagorean law:
BS=(a*(t*t)) / 2, where a is acceleration; t - time (a * t * t - this is the speed).
If we now use the Pythagorean formula, then:
R2+t2+v2 = R2+(a*t2*2*R) / 2+ (a*t2/2)2, where R is the radius and the alphanumeric spelling without the multiplication sign is the degree.
Huygens admitted that, since the time t is small, it can be ignored in the calculations. Having transformed the previous formula, she came to the well-known Acs = (v * v) / r.
However, since time is squared, a progression occurs: the larger t, the higher the error. For example, for 0.9, almost the total value of 20% is not taken into account.
The concept of centripetal acceleration is important for modern science, but it is obvious that it is too early to put an end to this issue.
Since the linear speed uniformly changes direction, then the movement along the circle cannot be called uniform, it is uniformly accelerated.
Angular velocity
Pick a point on the circle 1 . Let's build a radius. For a unit of time, the point will move to the point 2 . In this case, the radius describes the angle. The angular velocity is numerically equal to the angle of rotation of the radius per unit time.
Period and frequency
Rotation period T is the time it takes the body to make one revolution.
RPM is the number of revolutions per second.
The frequency and period are related by the relationship
Relationship with angular velocity
Line speed
Each point on the circle moves at some speed. This speed is called linear. The direction of the linear velocity vector always coincides with the tangent to the circle. For example, sparks from under a grinder move, repeating the direction of instantaneous speed.
Consider a point on a circle that makes one revolution, the time that is spent - this is the period T. The path traveled by a point is the circumference of a circle.
centripetal acceleration
When moving along a circle, the acceleration vector is always perpendicular to the velocity vector, directed to the center of the circle.
Using the previous formulas, we can derive the following relations
Points lying on the same straight line emanating from the center of the circle (for example, these can be points that lie on the wheel spoke) will have the same angular velocities, period and frequency. That is, they will rotate in the same way, but with different linear speeds. The farther the point is from the center, the faster it will move.
The law of addition of velocities is also valid for rotational motion. If the motion of a body or frame of reference is not uniform, then the law applies to instantaneous velocities. For example, the speed of a person walking along the edge of a rotating carousel is equal to the vector sum of the linear speed of rotation of the edge of the carousel and the speed of the person.
The Earth participates in two main rotational movements: daily (around its axis) and orbital (around the Sun). The period of rotation of the Earth around the Sun is 1 year or 365 days. The Earth rotates around its axis from west to east, the period of this rotation is 1 day or 24 hours. Latitude is the angle between the plane of the equator and the direction from the center of the Earth to a point on its surface.
According to Newton's second law, the cause of any acceleration is a force. If a moving body experiences centripetal acceleration, then the nature of the forces that cause this acceleration may be different. For example, if a body moves in a circle on a rope tied to it, then the acting force is the elastic force.
If a body lying on a disk rotates along with the disk around its axis, then such a force is the force of friction. If the force ceases to act, then the body will continue to move in a straight line
Consider the movement of a point on a circle from A to B. The linear velocity is equal to v A and v B respectively. Acceleration is the change in speed per unit of time. Let's find the difference of vectors.
Allows us to exist on this planet. How can you understand what constitutes centripetal acceleration? The definition of this physical quantity is presented below.
Observations
The simplest example of the acceleration of a body moving in a circle can be observed by rotating a stone on a rope. You pull the rope, and the rope pulls the rock towards the center. At each moment in time, the rope gives the stone a certain amount of movement, and each time in a new direction. You can imagine the movement of the rope as a series of weak jerks. A jerk - and the rope changes its direction, another jerk - another change, and so on in a circle. If you suddenly let go of the rope, the jerks will stop, and with them the change in direction of speed will stop. The stone will move in the direction tangent to the circle. The question arises: "With what acceleration will the body move at this instant?"
formula for centripetal acceleration
First of all, it is worth noting that the movement of the body in a circle is complex. The stone participates in two types of movement at the same time: under the action of a force, it moves towards the center of rotation, and at the same time, tangentially to the circle, it moves away from this center. According to Newton's Second Law, the force holding a stone on a string is directed toward the center of rotation along that string. The acceleration vector will also be directed there.
Let for some time t, our stone, moving uniformly at a speed V, gets from point A to point B. Suppose that at the moment when the body crossed point B, the centripetal force ceased to act on it. Then for a period of time it would hit the point K. It lies on the tangent. If at the same time only centripetal forces acted on the body, then in time t, moving with the same acceleration, it would end up at point O, which is located on a straight line representing the diameter of a circle. Both segments are vectors and obey the vector addition rule. As a result of the summation of these two movements for a period of time t, we obtain the resulting movement along the arc AB.
If the time interval t is taken negligibly small, then the arc AB will differ little from the chord AB. Thus, it is possible to replace movement along an arc with movement along a chord. In this case, the movement of the stone along the chord will obey the laws of rectilinear motion, that is, the distance AB traveled will be equal to the product of the speed of the stone and the time of its movement. AB = V x t.
Let us denote the desired centripetal acceleration by the letter a. Then the path traveled only under the action of centripetal acceleration can be calculated using the formula of uniformly accelerated motion:
Distance AB is equal to the product of speed and time, i.e. AB = V x t,
AO - calculated earlier using the uniformly accelerated motion formula for moving in a straight line: AO = at 2 / 2.
Substituting these data into the formula and transforming them, we get a simple and elegant formula for centripetal acceleration:
In words, this can be expressed as follows: the centripetal acceleration of a body moving in a circle is equal to the quotient of dividing the linear velocity squared by the radius of the circle along which the body rotates. The centripetal force in this case will look like the picture below.
Angular velocity
The angular velocity is equal to the linear velocity divided by the radius of the circle. The converse is also true: V = ωR, where ω is the angular velocity
If we substitute this value into the formula, we can get the expression for the centrifugal acceleration for the angular velocity. It will look like this:
Acceleration without speed change
And yet, why doesn't a body with acceleration directed towards the center move faster and move closer to the center of rotation? The answer lies in the wording of acceleration itself. The facts show that circular motion is real, but that it requires acceleration towards the center to maintain it. Under the action of the force caused by this acceleration, there is a change in the momentum, as a result of which the trajectory of motion is constantly curved, all the time changing the direction of the velocity vector, but not changing its absolute value. Moving in a circle, our long-suffering stone rushes inward, otherwise it would continue to move tangentially. Every moment of time, leaving on a tangent, the stone is attracted to the center, but does not fall into it. Another example of centripetal acceleration would be a water skier making small circles on the water. The figure of the athlete is tilted; he seems to be falling, continuing to move and leaning forward.
Thus, we can conclude that acceleration does not increase the speed of the body, since the velocity and acceleration vectors are perpendicular to each other. Added to the velocity vector, acceleration only changes the direction of motion and keeps the body in orbit.
Safety margin exceeded
In the previous experience, we were dealing with an ideal rope that did not break. But, let's say our rope is the most common, and you can even calculate the effort after which it will simply break. In order to calculate this force, it is enough to compare the safety margin of the rope with the load that it experiences during the rotation of the stone. By rotating the stone at a higher speed, you give it more movement, and therefore more acceleration.
With a jute rope diameter of about 20 mm, its tensile strength is about 26 kN. It is noteworthy that the length of the rope does not appear anywhere. Rotating a 1 kg load on a rope with a radius of 1 m, we can calculate that the linear speed required to break it is 26 x 10 3 = 1kg x V 2 / 1 m. Thus, the speed that is dangerous to exceed will be equal to √ 26 x 10 3 \u003d 161 m / s.
Gravity
When considering the experiment, we neglected the action of gravity, since at such high speeds its influence is negligibly small. But you can see that when unwinding a long rope, the body describes a more complex trajectory and gradually approaches the ground.
celestial bodies
If we transfer the laws of circular motion into space and apply them to the motion of celestial bodies, we can rediscover several long-familiar formulas. For example, the force with which a body is attracted to the Earth is known by the formula:
In our case, the factor g is the very centripetal acceleration that was derived from the previous formula. Only in this case, the role of a stone will be played by a celestial body attracted to the Earth, and the role of a rope will be the force of earth's attraction. The factor g will be expressed in terms of the radius of our planet and the speed of its rotation.
Results
The essence of centripetal acceleration is the hard and thankless work of keeping a moving body in orbit. A paradoxical case is observed when, with constant acceleration, the body does not change its velocity. To the untrained mind, such a statement is rather paradoxical. Nevertheless, when calculating the motion of an electron around the nucleus, and when calculating the speed of rotation of a star around a black hole, centripetal acceleration plays an important role.