Motion is a physical process that involves changing the spatial coordinates of a body. To describe the motion in physics, special quantities and concepts are used, the main of which is acceleration. In this article, we will examine the question that this is normal acceleration.
General definition
Acceleration in physics is understood as the speed of a change in speed. Speed itself is a vector kinematic characteristic. Therefore, in the definition of acceleration is meant not only a change in the absolute value, but also a change in the direction of speed. What does the formula look like? For full acceleration a¯, it is written in the following form:
a¯ = dv¯ / dt
That is, to calculate a¯, it is necessary to find the time derivative of the velocity vector with respect to time at a given moment. The formula shows that a¯ is measured in meters per second squared (m / s 2 ).
The direction of full acceleration a¯ is not connected in any way with the vector v¯. However, it coincides with the vector dv¯.
The reason for the appearance of acceleration in moving bodies is the external force of any nature acting on them. Acceleration never occurs if the external force is zero. The direction of action of the force coincides with the direction of acceleration a¯.
Curved path
In the general case, the considered quantity a¯ has two components: normal and tangent. But first of all, recall what a trajectory is. In physics, a trajectory is understood as a line along which a body passes a certain path in the process of movement. Since the trajectory can be either a straight line or a curve, the movement of bodies is divided into two types:
- straightforward;
- curvilinear.
In the first case, the body velocity vector can only change to the opposite. In the second case, the velocity vector and its absolute value are constantly changing.
As you know, the speed is directed tangent to the path. This fact allows us to introduce the following formula:
v¯ = v * u¯
Here u¯ is the unit tangent vector. Then the expression for full acceleration will be written in the form:
a¯ = dv¯ / dt = d (v * u¯) / dt = dv / dt * u¯ + v * du¯ / dt.
In obtaining the equality, we used the rule of calculating the derivative of the product of functions. Thus, the full acceleration a¯ is represented as the sum of two components. The first is its tangent component. This article is not considered. We only note that it characterizes the change in the velocity modulus v¯. The second term is normal acceleration. About him below in the article.
Normal point acceleration
We denote this acceleration component by a n ¯. Let's write an expression for it again:
a n ¯ = v * du¯ / dt
The normal acceleration equation a n ¯ can be written in explicit form if the following mathematical transformations are carried out:
a n ¯ = v * du¯ / dt = v * du¯ / dl * dl / dt = v 2 / r * r e ¯.
Here l is the path traveled by the body, r is the radius of curvature of the trajectory, r e ¯ is the unit radius vector that is directed to the center of curvature. This equality allows us to draw some important conclusions regarding the question that this is normal acceleration. Firstly, it does not depend on the change in the velocity modulus and is proportional to the absolute value of v¯, and secondly, it is directed to the center of curvature, that is, along the normal to the tangent at a given point on the trajectory. That is why the component a n ¯ is called normal or centripetal acceleration. Finally, thirdly, a n ¯ is inversely proportional to the radius of curvature r, which everyone felt experimentally on himself when he was a passenger in a car entering a lengthy and sharp turn.
Centripetal and centrifugal forces
It was noted above that the cause of any acceleration is the force. Since normal acceleration is a component of full acceleration that is directed toward the center of curvature of the trajectory, there must be some centripetal force. Its nature is most easily traced by various examples:
- Unwinding a stone tied to the end of a rope. In this case, the centripetal force is the tension of the rope.
- Long turn of the car. The centripetal force of friction of car tires on the road surface.
- The rotation of the planets around the sun. Gravitational attraction plays the role of the force in question.
In all these examples, the centripetal force leads to a change in the rectilinear trajectory. In turn, it is hindered by the inertial properties of the body. They are associated with centrifugal force. This force, acting on the body, is trying to "throw" it out of a curved path. For example, when a car makes a turn, passengers are pressed against one of the doors of the vehicle. This is the action of centrifugal force. She, in contrast to the centripetal, is fictitious.
Task example
As you know, our Earth rotates in a circular orbit around the Sun. It is necessary to determine the normal acceleration of the blue planet.
To solve the problem, we use the formula:
a n = v 2 / r.
From the reference data, we find that the linear velocity v of our planet is 29.78 km / s. The distance r to our star is 149 597 871 km. Translating these numbers into meters per second and meters, respectively, substituting them in the formula, we get the answer: a n = 0.006 m / s 2 , which is 0.06% of the value of the acceleration of gravity on the planet.