Solution to problem 15.7.4 from the collection of Kepe O.E.

Problem 15.7.4 is to determine the angular velocity of gears 1 and 2 after two revolutions, if they have the same mass of 2 kg and are driven by a constant moment of a pair of forces M = 1 N • m, and the radius of gyration of each wheel relative to the axis of rotation is 0 .2 m. The answer to the problem is 12.5.

This digital product is the solution to problem 15.7.4 from the collection of Kepe O.?. This collection is one of the most popular textbooks in physics and mathematics. Solving this problem will allow students to better understand the principles of rotation of a rigid body and apply them in practice.

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The digital product that I am describing is a solution to problem 15.7.4 from the collection of Kepe O.?. The task is to determine the angular velocity of gears 1 and 2 after two revolutions, if they have the same mass of 2 kg, are driven by a constant moment of a pair of forces M = 1 N • m, and the radius of gyration of each wheel relative to the axis of rotation is 0.2 m The solution to the problem is presented in a beautiful HTML format and contains all the necessary calculations and explanations. This product will be useful to students who want to better understand the principles of rotation of a rigid body and apply them in practice. Presentation in a structured and easily accessible format ensures ease of use and enhances aesthetic appeal. This digital product will help pupils and students improve their knowledge in the field of physics and mathematics and prepare for exams and testing. The answer to the problem is 12.5.

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Solution to problem 15.7.4 from the collection of Kepe O.?. is associated with determining the angular velocity of the wheels after two revolutions, if the mass and radius of inertia of each of the wheels are given, as well as the constant moment of a pair of forces that sets the wheels in motion from a state of rest.

To solve the problem, it is necessary to use the law of conservation of energy of rotational motion. According to this law, the change in the kinetic energy of rotational motion is equal to the work of external forces applied to the system. In this case, the external force is a pair of forces that creates a torque.

Thus, we can write the equation:

ΔE = A,

where ΔE is the change in the kinetic energy of rotation of the wheels, and A is the work done by a pair of forces during two revolutions.

It is known that when wheels rotate, their kinetic energy is determined by the formula:

E = (Iω²)/2,

where I is the moment of inertia of the wheel relative to the axis of rotation, and ω is the angular velocity of the wheel.

Therefore, to change the kinetic energy of the wheels we can write:

ΔE = E2 - E1 = (Iω2² - Iω1²)/2,

where E1 and E2 are the kinetic energy of the wheels at the beginning and end of movement, respectively.

The work done by a pair of forces during two revolutions is equal to:

A = МΔφ = 2πМ,

where M is the constant moment of the pair of forces, and Δφ = 2π is the full angle of rotation of the wheels for two revolutions.

Now you can substitute the known values ​​into the equation ΔE = A:

(Iω2² - Iω1²)/2 = 2πM,

and solve it relative to the angular velocity ω2:

ω2 = sqrt(2πМ/I) + ω1,

where ω1 is the initial angular velocity of the wheels, which is zero, since the wheels are at rest.

Thus, to find the angular velocity of the wheels after two revolutions, you need to substitute the known values ​​into the formula and solve it:

ω2 = sqrt(2π * 1 N•m / (2 * 0.2 m² * 2 kg)) = 12.5 rad/s.

Answer: 12.5 rad/s.

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