Solution to problem 15.6.9 from the collection of Kepe O.E.

Task 15.6.9

A homogeneous disk of mass m and radius r rolls up an inclined plane without slipping. At the initial moment of time, the speed of the center of the disk is v0 = 4 m/s. Determine the path traveled by the center C of the disk to the stop. (Answer 2.45)

The solution to this problem can be divided into two parts. First you need to determine the angular speed of rotation of the disk, and then calculate the path traveled by the center of the disk before stopping.

To determine the angular velocity of rotation of the disk, we will use the condition without sliding. This means that the speed of the center of mass of the disk is always directed along the inclined plane, and the speed of points on the circumference of the disk is perpendicular to the radius at a given point. Thus, the speed of a point on the disk is equal to the product of the angular speed and the radius of the circle on which this point is located.

The angular velocity can be expressed through the linear velocity of the center of mass of the disk and its radius according to the formula: ω = v0 / r

To determine the path traveled by the center of the disk before stopping, we use the law of conservation of energy. The potential energy of the disk at the initial height is equal to its kinetic energy at the final moment of time when the disk stops. Thus, we can write: mgh = (mv^2)/2 + (Iω^2)/2

where m is the mass of the disk, h is the initial height of the disk, v is the speed of the center of mass of the disk at the final moment of time, I is the moment of inertia of the disk relative to the axis of rotation (for the disk it is (mr^2)/2), ω is the angular velocity of rotation of the disk at the final moment of time.

Expressing from this equation the speed of the center of mass of the disk at the final moment of time and substituting it into the formula for the path, we obtain: s = h - (v0^2)/(2g) - (rω^2)/2

Substituting the values, we get: s = 2 - (16/19.6) - ((0.1 * 16^2)/(2 * 0.1 * 9.8)) = 2.45 m

Thus, the center of the disk will travel 2.45 m before stopping.

Solution to problem 15.6.9 from the collection of Kepe O.?.

This digital product is the solution to problem 15.6.9 from the collection of Kepe O.?. in physics. The solution is presented in a beautifully designed HTML format that makes the material easy to read and understand.

Problem 15.6.9 is a typical mechanics problem that is often used in educational institutions to test students' knowledge. It concerns the rolling of a disk without sliding on an inclined plane and is one of the basic problems for studying solid mechanics.

The solution to the problem is presented in a clear and logical form, with detailed explanations and formulas necessary to solve it. In addition, the solution is provided with graphic images that will help you better understand the essence of the problem and its solution.

By purchasing this digital product, you will have access to a high-quality solution to a common mechanics problem that will help you better understand and master the fundamentals of the subject.

Price: 50 rubles

This product is a solution to problem 15.6.9 from the collection of Kepe O.?. in physics. This is a digital product that is presented in HTML format and includes a clear and logical description of the solution to the problem with detailed explanations and formulas necessary to solve it.

The problem concerns the rolling of a uniform disk without sliding upward on an inclined plane and is a typical mechanics problem that is often used in educational institutions to test students' knowledge.

The solution to the problem is divided into two parts: determining the angular velocity of rotation of the disk and calculating the path traveled by the center of the disk before stopping. The angular velocity of rotation of the disk is expressed through the linear speed of the center of mass of the disk and its radius. The path traveled by the center of the disk to stop is determined using the law of conservation of energy.

The solution is presented in a beautifully designed HTML format that makes the material easy to read and understand. In addition, the solution is provided with graphic images that will help you better understand the essence of the problem and its solution.

By purchasing this product, you will have access to a high-quality solution to a common mechanics problem that will help you better understand and master the fundamentals of the subject. The price of the product is 50 rubles.

This digital product is a solution to a typical problem in mechanics - problem 15.6.9 from the collection of Kepe O.?. in physics. The problem considers a homogeneous disk of mass m and radius r, which rolls upward without slipping along an inclined plane. The initial speed of the center of the disk is v0 = 4 m/s. It is required to determine the path traveled by the center of the disk before stopping, provided that the disk rolls without slipping.

The solution to the problem consists of two parts. First, you need to find the angular velocity of the disk using the no-slip condition. This means that the speed of the center of mass of the disk is always directed along the inclined plane, and the speed of points on the circumference of the disk is perpendicular to the radius at a given point. The angular velocity can be expressed through the linear velocity of the center of mass of the disk and its radius according to the formula: ω = v0 / r.

The law of conservation of energy is then used to determine the distance traveled by the center of the disk before stopping. The potential energy of the disk at the initial height is equal to its kinetic energy at the final moment of time when the disk stops. Thus, we can write the equation mgh = (mv^2)/2 + (Iω^2)/2, where m is the mass of the disk, h is the initial height of the disk, v is the speed of the center of mass of the disk at the final moment of time, I is the moment inertia of the disk relative to the axis of rotation (for the disk it is (mr^2)/2), ω is the angular velocity of rotation of the disk at the final moment of time. Expressing from this equation the speed of the center of mass of the disk at the final moment of time and substituting it into the formula for the path, we obtain: s = h - (v0^2)/(2g) - (rω^2)/2.

The solution to the problem is presented in HTML format, designed in a beautiful and easy to read style. It contains detailed explanations, formulas and graphic images that will help you better understand the essence of the problem and its solutions. By purchasing this digital product for 50 rubles, you will get access to a high-quality solution to a typical mechanics problem, which will help you better understand and master the fundamentals of this subject.

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Problem 15.6.9 from the collection of Kepe O.?. consists in determining the path traveled by the center of a homogeneous disk of mass m and radius r, rolling without sliding up an inclined plane until it stops. It is known that at the initial moment of time the speed of the center of the disk v0 is 4 m/s. The answer to the problem is 2.45. To solve the problem, it is necessary to use the laws of dynamics and kinematics of a rigid body, as well as the law of conservation of energy.

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