In physics, there are several types of acceleration that are used to describe one type or another of the mechanical movement of bodies in space. All of these species are vector quantities. In this article we will not consider the question of where acceleration is directed, but we will focus on the formulas of the acceleration module.
What is acceleration?
The most complete definition of this kinematic characteristic can be given as follows: acceleration is a value that shows the speed of change of speed over time. It is about changing both the module and the direction. Mathematically, acceleration is calculated as follows:
a = dv / dt.
It is called instantaneous, that is, fair for a particular moment in time t. To find the average value of the acceleration modulus, this formula must be used:
a = (v 2 - v 1 ) / (t 2 - t 1 ).
Where v 2 and v 1 are the speeds at time t 2 and t 1, respectively.
The units of measurement of the physical quantity under study are meters per square second (m / s 2 ). Many may be embarrassed by raising to the second power units of time, however, it is not difficult to understand the meaning of the unit m / s 2 if it is presented in the form [m / s] / s. The last entry means a change in speed by one unit for one unit of time.
Straight and acceleration
The simplest path for moving bodies in space is a straight line. If the speed during movement along such a trajectory does not change, then there is no need to talk about acceleration, since it will be zero.
A straight-line, equally accelerated (equally slow) movement is widespread in technology. For example, when starting a car or when braking it, we have this particular type of movement. For its mathematical description, use the following equalities:
v = v 0 ± a * t;
l = v 0 * t ± a * t 2/2.
Here v 0 is a certain initial velocity of the body, which can also be equal to zero, l is the path traveled by the body to time t. The + sign indicates the acceleration of the body, the - sign indicates its inhibition. It is important to remember that the time t when using the written formulas begins to be counted from the moment a constant acceleration a appears in the body. Taking into account the written equalities, the formulas of the body acceleration module take the form:
± a = (v - v 0 ) / t;
± a = 2 * (l - v 0 * t) / t 2 .
As a rule, if the body accelerates, then they speak of positive acceleration, if it slows down its movement, then they speak of a negative value of a. It is easy to verify that both formulas lead to the same unit of acceleration (m / s 2 ).
Full acceleration and its components when the body moves along a curve
In the case of a body moving along a curved path, it is convenient to represent the value of a in the form of two mutually perpendicular components. They are called tangential a t and normal a n accelerations. For this case, the formula of the point acceleration module takes the form:
a = √ (a t 2 + a n 2 ).
The tangential component should be calculated through the time derivative of the function v (t). The normal component is determined not by a change in the modulus of speed, but by its magnitude. For its calculation, use the following expression:
a n = v 2 / r.
Here r is the radius of curvature of the trajectory, which in the case of rotation around the circle coincides with the radius of the latter.
For completeness of information, we note that the curvilinearity of the body trajectory is a sufficient sign of the presence of a nonzero normal component of acceleration. Moreover, the value of a t can be equal to zero, which is valid for uniform rotation of bodies.
Angular acceleration
As noted in the introduction, there are several types of acceleration. One of them is the angular kinematic value. Denote it by α. By analogy with linear acceleration, the formula of the angular acceleration module has the form:
α = dω / dt.
Where the Greek letter ω (omega) denotes the angular velocity, the units of which are radians per second. The value of α shows how quickly the body increases or slows down its rotation speed.
The angular acceleration can be associated with a linear quantity. This is done using the following formula:
α = a t / r.
It is important to understand that angular acceleration is a convenient way of representing the tangential component of full acceleration in the case of rotational motion. The convenience here lies in the independence of the quantity α from the distance to the axis of rotation r. In turn, the component a t increases linearly with increasing radius of curvature r.
Problem solving example
It is known that the body rotates in a circle whose radius is 0.2 meters. The rotation is accelerated, while the speed changes in time according to the following law:
v = 2 + 3 * t 2 + 2 * t 3 .
It is necessary to determine the tangential, normal, full and angular acceleration at time 3 seconds.
We will begin to solve this problem in order. The tangential component is determined through the derivative of velocity. We have:
a t = dv / dt = 6 * t + 6 * t 2 = 6 * 3 + 6 * 9 = 76 m / s 2 .
Note that this is a very large acceleration compared to the acceleration of gravity (9.81 m / s 2 ).
The normal component is calculated as follows:
a n = v 2 / r = 1 / r * (2 + 3 * t 2 + 2 * t 3 ) 2 = 1 / 0.2 * (2 + 27 + 54) 2 = 34445 m / s 2 .
Now you can calculate the full acceleration. It will be equal to:
a = √ (a t 2 + a n 2 ) = √ (76 2 + 34445 2 ) = 34445.1 m / s 2 .
That is, the full acceleration is almost completely formed by the normal component.
Finally, the angular acceleration is determined by the formula:
α = a t / r = 76 / 0.2 = 380 rad / s 2 .
The obtained value corresponds to an increase in the angular velocity by approximately 60 revolutions per second.