Centripetal acceleration - derivation of the formula and practical application
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 stone to the center. At each moment of 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 in the form of 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 along with them the change in the direction of speed will stop. The stone will move in the direction of the tangent to the circle. The question arises: "With what acceleration will the body move at this instant?"
Centripetal acceleration formula
First of all, it is worth noting that the movement of the body in a circle is difficult.A stone participates in two types of movement at the same time: under the action of 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 the stone on the rope is directed to the center of rotation along this rope. There will also be directed to the acceleration vector.
Let for some time t our stone, moving uniformly at speed V, falls from point A to point B. Suppose that at the time when the body crossed point B, a centripetal force ceased to act on it. Then for a period of time it would fall at the point K. It lies on the tangent. If at the same moment of time only centripetal forces would act on the body, then in time t, moving with the same acceleration, it would appear at the point O, which is located on a straight line, which represents the diameter of a circle. Both segments are vectors and obey the vector addition rule. As a result of the summation of these two motions in the time interval t, we obtain the resulting motion along the arc AB.
If the time interval t is taken to be negligible, then the arc AB will differ little from the chord AB.Thus, it is possible to replace movement along an arc by 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 traveled AB will be equal to the product of the speed of the stone at the time of its movement. AB = V x t.
Denote the desired centripetal acceleration by the letter a. Then the path covered only under the action of centripetal acceleration can be calculated using the uniformly accelerated motion formula:
AO = at2/ 2.
The distance AB is equal to the product of speed and time, that is, AB = V x t,
AO - previously calculated by the formula of uniformly accelerated motion to move in a straight line: AO = at2/ 2.
Substituting this data into a formula and transforming it, we get a simple and elegant centripetal acceleration formula:
a = v2/ R
This can be expressed in words as follows: the centripetal acceleration of a body moving in a circle is equal to the quotient from dividing the linear velocity in the square by the radius of the circle in which the body rotates. The centripetal force in this case will look like the image below.
The angular velocity is equal to the quotient of the linear velocity divided by the radius of the circle.The converse is also true: V = ωR, where ω is the angular velocity
If you substitute this value in the formula, you can get the expression of centrifugal acceleration for the angular velocity. It will look like this:
a = ω2R.
Acceleration without changing speed
And yet, why does a body with acceleration directed toward the center not move faster and does not move closer to the center of rotation? The answer lies in the very formulation of acceleration. The facts show that driving in a circle is real, but in order to maintain it, acceleration directed towards the center is required. Under the action of a force caused by this acceleration, a change in the amount of motion occurs, as a result of which the trajectory of motion constantly bends, changing the direction of the velocity vector all the time, but not changing its absolute value. Moving in a circle, our long-suffering stone rushes inward, otherwise it would continue to move on a tangent. 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 can be a water skier, describing small circles on the water.Athlete's figure is tilted; he seems to fall, 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. Adding to the velocity vector, acceleration only changes the direction of motion and keeps the body in orbit.
Excess safety margin
In the previous experience we dealt with an ideal rope that was not torn. 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 sufficient to compare the safety margin of the rope with the load that it experiences in the process of the stone rotation. By rotating the stone at a faster speed, you give it a greater amount of movement, and therefore a greater acceleration.
With a diameter of jute rope of about 20 mm, its tensile strength is about 26 kN. It is noteworthy that the length of the rope does not appear anywhere. By rotating a load of 1 kg in size on a rope with a radius of 1 m, it can be calculated that the linear velocity required to break it is 26 x 103= 1kg x V2/ 1 m. Thus, the speed, which is dangerous to exceed, will be equal to √26 x 103= 161 m / s.
When considering the experience, we neglected the action of gravity, since at such high speeds its effect is negligible. But you can see that when unwinding a long rope, the body describes a more complex trajectory and gradually approaches the ground.
If you transfer the laws of motion in a circle to space and apply them to the motion of celestial bodies, you can rediscover a few long-known formulas. For example, the force with which a body is attracted to the Earth is known by the formula:
F = m * g.
In our case, the factor g is the very centripetal acceleration, which was derived from the previous formula. Only in this case, the role of a stone will be performed by a celestial body, attracted to the Earth, and the role of a rope is the force of gravity. The multiplier g will be expressed through the radius of our planet and the speed of its rotation.
The essence of centripetal acceleration is the hard and thankless job of keeping a moving body in orbit. There is a paradoxical case when, with constant acceleration, the body does not change the magnitude of its speed. For the untrained mind, such a statement is rather paradoxical.Nevertheless, in the calculation of the motion of an electron around the nucleus, and in the calculation of the speed of rotation of a star around a black hole, the centripetal acceleration plays not the last role.