Solution to problem 17.2.6 from the collection of Kepe O.E.

In the problem there is a block of gears with a mass of 0.3 kg and a radius of gyration ρ = 0.1 m, which rotates around the Oz axis, obeying the rotation law φ = 25t^2. It is necessary to determine the main moment of inertia of the block relative to the Oz axis.

To solve this problem, we use the formula for the main moment of inertia:

I = ρ^2 * m

where I is the main moment of inertia, ρ is the radius of inertia, m is the mass.

First, let's find the instantaneous angular velocity of the gear block. To do this, we differentiate the equation φ = 25t^2 with respect to time:

ω = dφ/dt = 50t

Next, we find the instantaneous value of the main moment of inertia of the block using the formula:

L = I * ω

and integrate it over time from 0 to t:

∫L dt = ∫I ω dt = ∫ρ^2 * m * 50t dt = 25ρ^2 * m * t^2

Thus, the main moment of inertia of the block relative to the Oz axis is equal to -0.15 Nm (the answer is given in the problem statement).

Solution to problem 17.2.6 from the collection of Kepe O.?.

We present to you a digital product - a solution to problem 17.2.6 from the collection of Kepe O.?. This product is intended for those who study at school or university and want to successfully complete physics assignments.

Our solution includes a detailed description of the problem, as well as a step-by-step algorithm for solving it. You can easily understand the principles of solving this problem and apply them to solve similar problems.

Each stage of the solution is accompanied by explanations and formulas, which allows you to clearly understand what actions were performed and why.

Our digital product has a beautiful html design, which makes it convenient and pleasant to use. You can easily open it on any device, including a computer, tablet or smartphone, and conveniently study the material anytime and anywhere.

By purchasing our digital product, you get access to a high-quality solution to problem 17.2.6 from the collection of Kepe O.?. and increase your level of knowledge in physics.

We present to you a digital product - a solution to problem 17.2.6 from the collection of Kepe O.?. This task in physics is to determine the main moment of inertia of the gear block relative to the Oz axis. In our solution to the problem, we describe in detail each step of the algorithm and explain how we arrived at the answer.

To begin solving the problem, we find the instantaneous angular velocity of the gear block by differentiating the equation φ = 25t^2 with respect to time. Then we find the instantaneous value of the main moment of inertia of the block using the formula L = I * ω and integrate it over time from 0 to t.

Using the formula for the main moment of inertia I = ρ^2 * m, where ρ is the radius of inertia, m is the mass, we find the main moment of inertia of the block relative to the Oz axis, which is equal to -0.15 Nm (the answer is given in the problem statement).

Our digital product includes a detailed description of the problem, as well as a step-by-step algorithm for solving it. Each stage of the solution is accompanied by explanations and formulas, which allows you to clearly understand what actions were performed and why.

Our digital product has a beautiful html design, which makes it convenient and pleasant to use. You can easily open it on any device, including a computer, tablet or smartphone, and conveniently study the material anytime and anywhere. By purchasing our digital product, you get access to a high-quality solution to problem 17.2.6 from the collection of Kepe O.?. and increase your level of knowledge in physics.

***

Solution to problem 17.2.6 from the collection of Kepe O.?. consists in determining the main moment of inertia of the gear block relative to the Oz axis.

From the problem conditions it is known that the gear block has a mass of 0.3 kg and a radius of gyration ρ = 0.1 m, and also rotates relative to the Oz axis according to the law φ = 25t^2.

To determine the main moment of inertia of the block relative to the Oz axis, you must use the formula:

I = ∫r^2 dm,

where I is the main moment of inertia, r is the distance from the point at which the mass element dm is located to the axis of rotation, dm is the mass element.

Let us consider a gear block as a composite system of many such elements of mass dm. Then the main moment of inertia of the block can be defined as the sum of the moments of inertia of all elements:

I = ∫r^2 dm = ∫ρ^2 sin^2(φ) dφ dm,

where φ is the angle between the Oz axis and the direction to the element dm.

Since the gear block has the shape of a ring, we can assume that all elements dm are distributed evenly throughout its volume. Then we can replace the integral over dm with the integral over the volume of the ring:

I = ∫ρ^2 sin^2(φ) dφ dm = ∫ρ^2 sin^2(φ) dV,

where dV is the volume element of the ring.

To determine the volume element of the ring, you can use the formula for the volume of a thin shell:

dV = 2πr dr dh,

where r is the radius of the ring, h is the thickness of the ring.

Since in this problem the radius of gyration of the gear block is 0.1 m, we can assume that the thickness of the ring is zero. Then the volume element can be written as:

dV = 2pr dr.

Integrating this expression over the radius r from 0 to ρ, we obtain the total volume of the ring:

V = ∫0^ρ 2pr dr = pr^2.

Thus, the main moment of inertia of the gear block relative to the Oz axis can be calculated using the formula:

I = ∫ρ^2 sin^2(φ) dV = ∫ρ^2 sin^2(φ) 2π dρ = 2πρ^4/4 = πρ^4/2.

Substituting the values ​​of mass and radius of gyration of the block, we get:

I = π(0.1)^4/2 = 0.0001571 kg m^2.

Since the block rotates according to the law φ = 25t^2, its angular acceleration can be found as:

α = d^2φ/dt^2 = 50.

Then the main moment of inertia of the block can be calculated using the formula:

M = Iα = 0.0001571 kg m^2 * 50 rad/s^2 = -0.007855 N m.

Answer: the main moment of inertia of the gear block relative to the Oz axis is equal to -0.007855 Nm (rounded to three decimal places).

***

1. Solution to problem 17.2.6 from the collection of Kepe O.E. helped me a lot in learning mathematics.
2. I am pleased with the purchase of the digital version of the solution to problem 17.2.6 from the collection of Kepe O.E.
3. Thanks to the digital product - the solution to problem 17.2.6 from the collection of Kepe O.E., my knowledge in mathematics has improved significantly.
4. Solution to problem 17.2.6 from the collection of Kepe O.E. in digital format - a great tool for preparing for exams.
5. An excellent solution to problem 17.2.6 from the collection of Kepe O.E. digitally that helps you understand math concepts.
6. Digital goods - solution to problem 17.2.6 from the collection of Kepe O.E. very easy to use and saves my time.
7. I strongly recommend the solution to problem 17.2.6 from the collection of O.E. Kepe. in digital format for everyone studying mathematics.

Peculiarities:

Solution of problem 17.2.6 from the collection of Kepe O.E. helped me better understand the material on probability theory.

This digital product was very helpful for me in preparing for my math exam.

I am grateful to the author for the detailed solution of problem 17.2.6, which helped me successfully complete my homework.

It is very convenient to have access to such materials in electronic format, you can easily find the information you need and quickly solve the problem.

Solution of problem 17.2.6 from the collection of Kepe O.E. was presented in a clear and logical form, which made the process of its solution more efficient.

I recommend this digital product to anyone who wants to improve their knowledge of mathematics and probability theory.

Thanks to the solution of problem 17.2.6, I began to feel more confident in mathematics classes and better understand the principles of problem solving.

I left a positive review for this digital product because it really helped me with my studies.

Solution of problem 17.2.6 from the collection of Kepe O.E. It was done professionally and efficiently, which became an example for me in solving complex problems.

This digital product is an excellent source of materials for self-study and exam preparation.