A quantitative study of the dynamics and kinematics of rotational motion requires knowledge of the moment of inertia of the material point and the solid relative to the axis of rotation. We consider in the article what parameter we are talking about, and also give a formula for its determination.
General physical quantity
First, we will determine the moment of inertia of a material point and a solid, and then we will show how it should be used in solving practical problems.
The indicated physical characteristic for a point having mass m, which rotates around an axis at a distance r, means the following quantity:
I = m * r².
It follows that the unit of measurement of the studied parameter is kilograms per square meter (kg * m²).
If instead of a point around the axis, a body of complex shape rotates, which has an arbitrary distribution of mass inside itself, then its moment of inertia is defined as follows:
I = ∫ m (r² * dm) = ρ * ∫ V (r² * dV).
Where ρ is the density of the body. Using the integral formula, one can determine the value of I for absolutely any rotation system.
The moment of inertia has exactly the same meaning for rotation as mass for translational motion. For example, everyone knows that a mop for mopping is easiest to rotate around an axis passing through its handle than through a perpendicular to it. This is due to the fact that the moment of inertia in the first case is much less than in the second.
The value of I for bodies of various shapes
When solving problems in physics for rotation, it is often necessary to know the moment of inertia for a body of a particular geometric shape, for example, for a cylinder, ball, or rod. If we apply the above formula for I, then it is easy to obtain the corresponding expression for all the marked bodies. The following are formulas for some of them:
rod: I = 1/12 * M * L²;
cylinder: I = 1/2 * M * R²;
scope: I = 2/5 * M * R².
Here I are given for the axis of rotation, which passes through the center of mass of the body. In the case of a cylinder, the axis is parallel to the generator of the figure. The moment of inertia for other geometric bodies and options for the location of the axis of rotation can be found in the corresponding tables. Note that to determine I different figures, it is enough to know only one geometric parameter and body weight.
Steiner theorem and formula
The moment of inertia can be determined if the axis of rotation is located at a certain distance from the body. To do this, one should know the length of this segment and the value I O of the body relative to the axis passing through the center of its mass, which should be parallel to the considered one. The connection between the parameter I O and the unknown value of I is fixed in the Steiner theorem. The moment of inertia of a material point and a solid is mathematically written as follows:
I = I O + M * h 2 .
Here M is the body mass, h is the distance from the center of mass to the axis of rotation, relative to which it is necessary to calculate I. This expression can be easily obtained independently if we use the integral formula for I and take into account that all points of the body are at distances r = r 0 + h .
Steiner's theorem greatly facilitates the definition of I for many practical situations. For example, if you need to find I for a rod of length L and mass M relative to the axis that passes through its end, then applying the Steiner theorem allows you to write:
I = I O + M * (L / 2) 2 = 1/12 * M * L 2 + M * L 2/4 = M * L 2/3.
You can turn to the corresponding table and see that it is precisely this formula that is given for a thin rod with an axis of rotation at its end.
Equation of moments
In rotation physics, there is a formula called the equation of moments. It looks like this:
M = I * α.
Here M is the moment of force, α is the angular acceleration. As you can see, the moment of inertia of the material point and the solid and the moment of force are linearly related to each other. The value of M determines the possibility of a certain force F to create a rotational motion with acceleration α in the system. To calculate M, use the following simple expression:
M = F * d.
Where d is the arm of the moment, which is equal to the distance from the force vector F to the axis of rotation. The smaller the arm d, the lesser the ability to create a rotation of the system.
The equation of moments in its meaning is fully consistent with Newton's second law. In this case, I plays the role of inertial mass.
Problem solving example
Imagine a system that is a cylinder mounted on a vertical axis using a weightless horizontal rod. It is known that the axis of rotation and the main axis of the cylinder are parallel to each other, and the distance between them is 30 cm. The mass of the cylinder is 1 kg and its radius is 5 cm. The figure is affected by a force of 10 N tangential to the rotation path, the vector of which passes through the main axis of the cylinder. It is necessary to determine the angular acceleration of the figure, which this force will cause.
First, we calculate the moment of inertia of cylinder I. To do this, apply the Steiner theorem, we have:
I = I O + M * d² = 1/2 * M * R² + M * d² = 1/2 * 1 * 0.05² + 1 * 0.3² = 0.09125 kg * m².
Before using the equation of moments, it is necessary to determine the moment of force M. In this case, we have:
M = F * d = 10 * 0.3 = 3 N * m.
Now you can determine the acceleration:
α = M / I = 3 / 0.09125 ≈ 32.9 rad / s².
The calculated angular acceleration indicates that every second the cylinder speed will increase by 5.2 revolutions per second.