The rotation of bodies is one of the important types of mechanical motion in technology and nature. Unlike linear displacement, it is described by its own set of kinematic characteristics. One of them is angular acceleration. We characterize this value in the article.
Rotation motion
Before talking about angular acceleration, we describe the type of motion to which it is applied. We are talking about rotation, which is the movement of bodies along circular paths. In order for the rotation to occur, some conditions must be met:
- the presence of an axis or point of rotation;
- the presence of a centripetal force that would keep the body in a circular orbit.
Examples of this type of movement are various attractions, such as a carousel. In technology, rotation manifests itself when the wheels and shafts move. In nature, the most striking example of this type of motion is the rotation of the planets around its own axis and around the Sun. The role of the centripetal force in these examples is played by the forces of interatomic interaction in solids and gravitational interaction.
Kinematic characteristics of rotation
These characteristics include three quantities: angular acceleration, angular velocity and angle of rotation. We will denote them by the Greek symbols α, ω, and θ, respectively.
Since the body moves in a circle, it is convenient to calculate the angle θ by which it will rotate in a certain time. This angle is expressed in radians (less often in degrees). Since the circle has a 2 × pi radian, we can write an equality connecting θ with the arc length L of the rotation:
L = θ × r
Where r is the radius of rotation. This formula is easy to obtain if we recall the corresponding expression for the circumference.
The angular velocity ω, like its linear counterpart, describes the speed of rotation around the axis, that is, it is determined according to the following expression:
ω¯ = d θ / dt
The quantity ω¯ is a vector. It is directed along the axis of rotation. The unit of measurement is radian per second (rad / s).
Finally, angular acceleration is a physical characteristic that determines the rate of change of ω¯, which is mathematically written as follows:
α¯ = d ω¯ / dt
The vector α¯ is directed towards the change in the velocity vector ω¯. It will be further said that angular acceleration is directed towards the vector of the moment of force. This value is measured in radians per square second (rad / s 2 ).
Moment of force and acceleration
If we recall Newton’s law, which connects force and linear acceleration into a single equality, then, transferring this law to the case of rotation, we can write the following expression:
M¯ = I × α¯
Here M¯ is the moment of force, which is the product of the force that tends to untwist the system, by the lever, the distance from the point of application of force to the axis. The value of I is an analog of body mass and is called the moment of inertia. The written formula is called the equation of moments. From it, the angular acceleration can be calculated as follows:
α¯ = M¯ / I
Since I is a scalar, α¯ is always directed towards the acting moment of force M¯. The direction of M¯ is determined by the rule of the right hand or the rule of the gimlet. The vectors M¯ and α¯ are perpendicular to the plane of rotation. The larger the moment of inertia the body has, the smaller the angular acceleration value can tell the system a fixed moment M¯.
Kinematic equations
To understand the important role angular acceleration plays in describing the motion of rotation, we write the formulas that relate the kinematic quantities studied above.
In the case of uniformly accelerated rotation, the following mathematical relations are valid:
ω = α × t;
θ = α × t 2/2
The first formula shows that the angular velocity will increase in time according to a linear law. The second expression allows you to calculate the angle by which the body will rotate in a known time t. The graph of the function θ (t) is a parabola. In both cases, angular acceleration is a constant.
If we use the relation formula between L and θ given at the beginning of the article, then we can obtain the expression for α through linear acceleration a:
α = a / r
If α is constant, then with increasing distance from the axis of rotation r, linear acceleration a will proportionally increase. That is why angular characteristics are used for rotation, unlike linear ones, they do not change with increasing or decreasing r.
Task example
The metal shaft, rotating at a speed of 2,000 revolutions per second, began to slow down its movement and after 1 minute completely stopped. It is necessary to calculate the angular acceleration of the shaft braking process. You should also calculate the number of revolutions that the shaft made before stopping.
The process of deceleration of rotation is described by the expression:
ω = ω 0 - α × t
The initial angular velocity ω 0 is determined through the frequency of rotation f in this way:
ω 0 = 2 × pi × f
Since we know the braking time, then we obtain the acceleration value α:
α = ω 0 / t = 2 × pi × f / t = 209.33 rad / s 2
This number should be taken with a minus sign, since we are talking about the braking of the system, and not about its acceleration.
To determine the number of revolutions that the shaft will make during braking, we apply the expression:
θ = ω 0 × t - α × t 2/2 = 376 806 rad.
The obtained value of the rotation angle θ in radians is simply converted into the number of revolutions made by the shaft until it stops completely using a simple division by 2 × pi:
n = θ / (2 × pi) = 60 001 revolutions.
Thus, we got all the answers to the questions of the problem: α = -209.33 rad / s 2 , n = 60 001 revolution.