Circular motion or rotational motion of solids is one of the important processes that are studied in the sections of physics - dynamics and kinematics. We will devote this article to the question of what measures the angular acceleration that appears during the rotation of bodies.
The concept of angular acceleration
Obviously, before giving an answer to the question of what angular acceleration is measured in physics, one should get acquainted with the concept itself.
In the mechanics of linear motion, acceleration plays the role of a measure of the speed of a change in speed and is introduced into physics through Newton’s second law. In the case of rotational motion, there is a quantity analogous to linear acceleration, which is called angular acceleration. The formula for its definition is written as:
α = dω / dt.
That is, the angular acceleration α is the first time derivative of the angular velocity ω. So, if the speed does not change during rotation, then the acceleration will be zero. If the speed linearly depends on time, for example, it increases constantly, then the acceleration α will take a constant nonzero positive value. A negative value of α indicates that the system slows down its rotation.
Rotation dynamics
In physics, any acceleration arises only when there is a nonzero external force acting on the body. In the case of a rotation motion, this force is replaced by a moment of force M equal to the product of the shoulder d by the force modulus F. The well-known equation of the moments of the dynamics of the rotational movement of bodies is written as
M = α * I.
Here I is the moment of inertia, playing the same role in the system as the mass during linear displacement. This formula allows you to calculate the value of α, as well as to determine in which the angular acceleration is measured. We have:
α = M / I = [N * m / (kg * m 2 )] = [N / (kg * m)].
We got the unit of measure α from the equation of moments, however, a newton is not a basic SI unit, so it should be replaced. To accomplish this task, we use Newton’s second law, we get:
1 N = 1 kg * m / s 2 ;
α = 1 [N / (kg * m)] = 1 kg * m / s 2 / (kg * m) = 1 [1 / s 2 ].
We got the answer to the question in which units angular acceleration is measured. It is measured in reverse square seconds. Second, in contrast to Newton, is one of the seven basic units of SI, therefore, the obtained unit for α is used in mathematical calculations.
The obtained unit of measure for angular acceleration is correct, however, it is difficult to understand the physical meaning of the quantity from it. In this regard, the problem can be solved in another way, using the physical definition of acceleration, which was recorded in the previous paragraph.
Angular speed and acceleration
Let us return to the definition of angular acceleration. In the kinematics of rotation, angular velocity determines the angle of rotation per unit time. As units for measuring the angle, you can use either degrees or radians. The latter are more often used. Thus, the angular velocity is measured in radians per second or abbreviated rad / s.
Since angular acceleration is a time derivative of ω, to obtain its units of measure, it is sufficient to divide the unit for ω per second. The latter means that the value of α will be measured in radians per square second (rad / s 2 ). So, 1 rad / s 2 means that for every second of rotation the angular velocity will increase by 1 rad / s.
The unit under consideration for α is similar to that obtained in the previous paragraph of the article, where the radian value was omitted, since it is implied in accordance with the physical meaning of angular acceleration.
Angular and centripetal acceleration
After answering the question of what is measured by angular acceleration (the formulas are given in the article), it is also useful to understand how it is related to centripetal acceleration, which is an integral characteristic of any rotation. The answer to this question sounds simple: angular and centripetal accelerations are completely different quantities that are independent.
Centripetal acceleration provides only a curvature of the body trajectory during rotation, while angular acceleration leads to a change in linear and angular velocities. So, in the case of uniform motion around the circumference, the angular acceleration is equal to zero, while the centripetal acceleration has a certain constant positive value.
The angular acceleration α is related to the linear tangent acceleration a by the following formula:
α = a / r.
Where r is the radius of the circle. Substituting the units of measure for a and r into this expression, we also get the answer to the question of what is measured by angular acceleration.
The solution of the problem
We solve the following problem from physics. A force of 15 N is tangent to a circle and acts on a material point. Knowing that this point has a mass of 3 kg and rotates around an axis with a radius of 2 meters, it is necessary to determine its angular acceleration.
This problem is solved using the equation of moments. The moment of force in this case is equal to:
M = F * r = 15 * 2 = 30 N * m.
The moment of inertia of a point is calculated by the following formula:
I = m * r 2 = 3 * 2 2 = 12 kg * m 2 .
Then the acceleration value will be equal to:
α = M / I = 30/12 = 2.5 rad / s 2 .
Thus, for every second the motion of a material point, its rotation speed will increase by 2.5 radians per second.