All types of movement that a person encounters throughout his life are studied in physics in the framework of the laws of classical Newtonian mechanics. This article will discuss the rotation of solids, the equations that describe it, and the formulas of angular acceleration.
Rotation in nature and technology
Before studying angular acceleration and the formulas that allow it to be determined, we consider what constitutes a rotation process.
In physics, rotation is understood as a type of motion in which each point of a body of arbitrary shape or a material point moves along a circular path. The axis of rotation passes through the center of this trajectory perpendicular to its plane . The considered type of displacement in space can occur with a constant speed or with a variable. In the latter case, it is said that the body has acceleration.
Examples of rotation in everyday life are the movement of the fan blades, the rotation of the wheels of a bicycle and a car, the rotation of the knives of a blender or the working part of a mixer, the movement of shafts and gears. In nature, you can also observe this type of movement, for example, the movement of our Earth around the Sun and its daily rotation around its axis.
What quantities describe the rotation?
Since the type of motion under consideration occurs along a circular path, it turns out to be convenient to describe it using quantities that use the value of the angle of the circular sector. We list these quantities.
Angle θ is the central angle of the circular sector. For clarity, it is shown below in the figure.
Here, the arc AB is the distance that the rotating body has traveled over a certain period of time t. Using the appropriate proportion, we can obtain a relationship between the length of this arc L and the angle θ expressed in radians. This relationship has the following mathematical form:
L = θ × r
The symbol r denotes the radius of the circle. Recall that one revolution along a circular path corresponds to the angle θ = 2 × pi.
The next important kinematic quantity of rotation is speed. It is usually denoted by the Greek letter ω (omega) and is determined in radians per second. The difference between this speed, which is called angular, from its linear counterpart is obvious, since the latter is measured in meters per second. Mathematically, the value of ω is determined as follows:
ω = dθ / dt
That is, the angular velocity shows the speed of rotation of the body. The recorded expression is called the instantaneous speed calculated in an infinitely small period of time (t; t + dt). The formula for the average speed ω m is written in the form:
ω m = θ / t
Moreover, on the angular gap θ, the instantaneous velocity ω can vary significantly.
Finally, the third important quantity that is used to describe non-uniform circular motion is the angular acceleration α. This value shows how the speed changes over a given period of time. The formula for angular acceleration through angular velocity is as follows:
α = dω / dt
It follows that the unit of measurement α is the radian per square second (rad / s 2 ).
Rotational dynamics and angular acceleration
Recalling Newton's laws for motion, we can say that any change in the mechanical displacement of a rigid body is associated with the action of some external force. In the case of rotation, it is not the force itself that is considered, but its moment. The latter is equal to the vector product of force by the vector of the distance from the axis to the point of application of force, that is:
M¯ = [r¯ × F¯]
This moment leads either to inhibition of rotation or to its acceleration. In both cases, the equation of motion in scalar form is written as:
M = I × α
Where I is the moment of inertia (analog of body mass for rectilinear motion). The last equality allows us to write a formula for calculating the angular acceleration in dynamics:
α = M / I.
Thus, knowing the moments of force and inertia, accelerations α can be calculated.
Kinematics of accelerated rotation
If the rotation speed does not change its value for some time, then there is no need to talk about the angular acceleration of the body, since it is equal to zero. If the speed changes over time, then the movement is called accelerated. The formula is valid for him:
ω = α × t
This equality is valid only when α is not a function of time, i.e., α = const. Such a movement is called uniformly accelerated.
The rotation angle for uniformly accelerated motion of the body from a place, that is, in the absence of an initial rotation speed, can be calculated as follows:
θ = α × t 2/2
From here we obtain the formula of angular acceleration through the central angle of rotation:
α = (2 × θ) / t 2
Note that if the body first rotated without acceleration at a speed of ω 0 , and then began to move accelerated, then for the angular distance traveled by it, we can write:
θ = ω 0 × t + α × t 2/2
From this it is easy to obtain the corresponding expression for α.
The relationship between linear and angular acceleration
Linear acceleration a shows the increase in speed v over time t. This value is measured in meters per second square, that is:
a = dv / dt
To find the relationship between the values of a and α, we turn to the following formula given earlier:
L = θ × r
We take the derivative of the left and right sides of the equation with respect to time, we obtain:
dL / dt = dθ / dt × r =>
v = ω × r
Take the dt derivative again:
dv / dt = dω / dt × r =>
a = α × r
We have obtained the relation between linear and angular accelerations. The expression shows that with constant acceleration α, the value of a will increase linearly with increasing radius of rotation.
The solution of the problem
It is known that a car moves with linear acceleration of 2 m / s 2 . It is necessary to determine the angular acceleration of its wheel if the wheel diameter is 40 cm.
The solution to this problem can be performed by conducting sequential mathematical calculations, and moving from linear to angular characteristics. However, the equality connecting the values of a and α was given above. It allows you to write the formula for the angular acceleration of the wheel. We have:
a = α × r =>
α = 2 × a / d, where d = 2 × r.
We substitute the data from the condition, write the answer: α = 10 rad / s 2 .