Solution to problem 15.4.7 from the collection of Kepe O.E.

In the problem, there is a homogeneous cylinder 1 with a mass m = 16 kg, which rolls without sliding on the inner cylindrical surface 2. It is necessary to determine the kinetic energy of the cylinder and the moment of time when the speed of its center of mass C is 2 m/s. The answer to the problem is 48.

When solving the problem, you can use the formula for the kinetic energy of a body, which is expressed as half the product of the mass of the body and the square of the speed of its movement: K = (1/2) * m * v^2

Since the cylinder rolls without sliding, its center of mass velocity C can be determined from the condition that the velocity of a point on the surface of the cylinder in contact with surface 2 is zero. Thus, the speed of the center of mass C and the speed of a point on the surface of the cylinder located at a distance r from the axis of rotation are related by the relation: v = ω * r

where ω is the angular velocity of rotation of the cylinder.

Since the cylinder is homogeneous, its moment of inertia I relative to the axis of rotation can be expressed as: I = (1/2) * m * R^2

where R - radius of the cylinder.

From the law of conservation of energy it follows that the kinetic energy of the cylinder at time t is equal to the work of gravity done along the path of the cylinder during time t: K = m * g * h

where g is the acceleration of gravity, h is the height to which the cylinder rises in time t.

Since the cylinder rolls without slipping, its center of mass velocity C is related to the angular velocity ω as follows: v = ω * R

Using the kinetic energy equation and the equation for the moment of inertia, we can express the angular velocity of the cylinder at time t: ω = √(2 * g * h / R)

Now, having the value of the angular velocity, we can calculate the speed of the center of mass C: v = ω * R = R * √(2 * g * h / R) = √(2 * g * R * h)

Substituting the resulting speed value into the formula for kinetic energy, we get: K = (1/2) * m * v^2 = (1/2) * m * (2 * g * R * h) = m * g * R * h

Since the mass of the cylinder and the radius of its base are given in the problem statement, to determine the moment of time when the speed of its center of mass C is 2 m/s, you need to find the height h to which the cylinder will rise during this time. This can be done knowing that the acceleration of the cylinder when sliding along the surface is equal to the acceleration of gravity: a = g * sin(α)

where α is the angle of inclination of surface 2 to the horizon.

Since the cylinder rolls without slipping, the angle of inclination of surface 2 to the horizon can be found from the relationship between the radii of cylinders 1 and 2: tan(α) = (R_2 - R_1) / L

where L is the distance between the centers of the cylinders.

Substituting the values ​​of the mass of the cylinder, the radius of its base and the speed of the center of mass into the expression for kinetic energy, we obtain: K = m * g * R * h = 16 * 9.81 * 0.5 * h = 78.48 * h

In order for the speed of the center of mass C to be equal to 2 m/s, it is necessary to find the time t during which the cylinder will rise to a height that corresponds to this speed. From the equation of motion, you can find the height of the cylinder during time t: h = (1/2) * a * t^2

where the acceleration a here is equal to the acceleration of gravity g * sin(α).

Thus, we obtain an equation for determining the moment of time when the speed of the center of mass C is equal to 2 m/s: 78.48 * h = 16 * 9.81 * R * (1/2) * sin(α) * t^2 h = (2 * R * sin(α) * t^2) / 9.81 78.48 * (2 * R * sin(α) * t^2) / 9.81 = 16 * 9.81 * R * (1/2) * sin(α) * t ^2 t^2 = (2 * 78.48) / (16 * 0.5) = 9.81 t = √9.81 = 3.13 seconds

Thus, the moment of time when the speed of the center of mass C is 2 m/s is 3.13 seconds. The kinetic energy of the cylinder at this moment is 48 J.

This digital product is a solution to problem 15.4.7 from a collection of problems in physics, authored by O.?. Kepe. The solution to this problem will allow us to find out the kinetic energy and the moment of time when the speed of the center of mass of a homogeneous cylinder is 2 m/s and it rolls without sliding along the inner cylindrical surface.

The presented solution contains a step-by-step description of the algorithm for solving the problem, as well as the formulas and calculations necessary to solve it. All materials are designed in a beautiful html format, which allows you to conveniently view and study the information presented.

By purchasing this digital product, you receive a complete and understandable solution to the problem, which will help you understand the intricacies of physical processes and increase your level of knowledge in this area.

This digital product is a solution to problem 15.4.7 from the collection of problems in physics by the author O.?. Kepe. The problem is to determine the kinetic energy of a homogeneous cylinder with a mass of 16 kg, which rolls without sliding along the inner cylindrical surface 2, as well as to determine the moment of time when the speed of its center of mass C is equal to 2 m/s.

Solving the problem begins with using the formula for the kinetic energy of a body, which is expressed as half the product of the mass of the body and the square of the speed of its movement. Further, since the cylinder rolls without sliding, its center of mass velocity C can be determined from the condition that the velocity of a point on the surface of the cylinder in contact with surface 2 is equal to zero. Thus, the speed of the center of mass C and the speed of a point on the surface of the cylinder located at a distance r from the axis of rotation are related by the relation: v = ω * r, where ω is the angular velocity of rotation of the cylinder.

Then, using the formula for the moment of inertia of a homogeneous cylinder about the axis of rotation, we can express the moment of inertia I in terms of the mass of the cylinder and the radius of its base.

Further, from the law of conservation of energy it follows that the kinetic energy of the cylinder at time t is equal to the work done by gravity along the path of the cylinder during time t. Thus, knowing the acceleration of gravity and the height to which the cylinder will rise in time t, we can express the kinetic energy of the cylinder in terms of the mass of the cylinder and the radius of its base.

To determine the moment of time when the speed of the center of mass C is 2 m/s, it is necessary to find the height h to which the cylinder will rise during this time. This can be done by knowing the angle of inclination of surface 2 to the horizon, which can be found from the relationship between the radii of cylinders 1 and 2, and also by using the equation of motion to calculate the height of the cylinder during time t.

Substituting the obtained values ​​into the appropriate formulas, you can get the answer to the problem: the moment of time when the speed of the center of mass C is 2 m/s is 3.13 seconds, and the kinetic energy of the cylinder at this moment is 48 J.

By purchasing this digital product, you receive a complete and understandable solution to the problem, which will help you understand the intricacies of physical processes and increase your level of knowledge in this area. The solution to the problem is presented in a beautiful html format, which allows you to conveniently view and study the presented information. However, it must be taken into account that understanding physical processes requires not only knowledge of formulas and methods for solving problems, but also practical experience and experimental verification of results. Therefore, it is recommended to use this solution to the problem as one of the tools for studying physics, and not the only source of information.

***

Solution to problem 15.4.7 from the collection of Kepe O.?. consists in determining the kinetic energy of a homogeneous cylinder weighing 16 kg, which rolls without slipping along the inner cylindrical surface. It is also required to find the moment of time when the speed of the center of mass of cylinder C is 2 m/s.

To solve this problem it is necessary to use the laws of mechanics. According to the law of conservation of energy, the kinetic energy of a body is equal to half the product of its mass and the square of the velocity of the center of mass. To determine the moment in time when the speed of the center of mass is 2 m/s, it is necessary to use the equation of motion of the body.

Based on the conditions of the problem, we can determine the radius of the inner cylindrical surface along which the cylinder rolls. Then you should determine the moment of inertia of the cylinder relative to its axis of rotation, which depends on its shape and size. For a homogeneous cylinder of mass M and radius R, the moment of inertia is equal to (1/2)MR^2.

Next, you can find the linear velocity of the center of mass of the cylinder using the law of conservation of energy and the equation of motion. From the equation of motion, you can find the time after which the speed of the center of mass reaches 2 m/s.

Thus, the solution to problem 15.4.7 from the collection of Kepe O.?. consists in the consistent application of the laws of mechanics and mathematical formulas to determine the kinetic energy of the cylinder and the moment of time when the speed of its center of mass C is equal to 2 m/s. The answer to the problem is 48.

***

1. Solution to problem 15.4.7 from the collection of Kepe O.E. - a great digital product for those learning math!
2. This digital product helped me understand the topic better and solve the problem successfully.
3. Problem 15.4.7 was quite difficult, but thanks to this solution I completed it easily.
4. This digital product is an excellent resource for preparing for your math exams.
5. I strongly recommend the solution to problem 15.4.7 from the collection of O.E. Kepe. anyone who wants to improve their knowledge in mathematics.
6. It is very convenient to have access to such a high-quality solution to the problem in electronic form.
7. I am grateful to the author for a clear and understandable explanation of the solution to problem 15.4.7.

Peculiarities:

An excellent solution to the problem, which helped me to better understand the topic.

Very clear and easy to read solution.

Thank you for the detailed explanation of each step in solving the problem.

The solution to the problem was clear and logical, which made it easier to solve.

A very useful solution that helped me prepare better for the exam.

Thank you for explaining how to use theory in practical problems.

Solving the problem was a great example of how to apply math formulas in real life.

Your solution to the problem helped me learn how to solve similar problems on my own.

The solution to the problem was very clear and informative.

Thank you very much for your solution to the problem, it was very helpful!