Solution to problem 14.2.18 from the collection of Kepe O.E.

14.2.18 Pulley 2 of radius R = 0.2 m, rotating with angular velocity ? = 20 rad/s, lifts homogeneous cylinder 1 with mass m = 50 kg. It is necessary to find the modulus of momentum of cylinder 1. (Answer 100)

The task is to find the modulus of momentum of cylinder 1, which is lifted by a homogeneous cylinder 2 of radius R = 0.2 m at the angular speed of rotation of the pulley ? = 20 rad/s. The mass of the cylinder is m = 50 kg.

To solve the problem, you can use the law of conservation of momentum, which is formulated as follows: the sum of the quantities of motion of all bodies in a closed system remains unchanged.

Thus, the modulus of momentum of cylinder 1 is equal to the modulus of momentum of the system of cylinder 1 and pulley 2 before the cylinder begins to rise. It is known that the quantities of motion of the pulley and the cylinder are equal in magnitude, that is:

p1 = p2

For a pulley, the amount of motion is:

p2 = I2 * w,

where I2 is the moment of inertia of the pulley, w is its angular velocity.

The moment of inertia of the pulley can be found using the formula:

I2 = 0.5 * M2 * R^2,

where M2 is the mass of the pulley, R is its radius.

Thus, the amount of movement of the pulley will be:

p2 = 0.5 * M2 * R^2 * w.

Similarly, for a cylinder the momentum can be written as:

p1 = m * v,

where v is the cylinder speed.

To find the speed of the cylinder, it is necessary to use the law of conservation of energy, which is formulated as follows: the total mechanical energy of a closed system remains unchanged.

Thus, the total mechanical energy of the system before the cylinder begins to rise is equal to the total mechanical energy of the system after the cylinder rises:

E1 + E2 = E1' + E2',

where E1 = m * g * h - potential energy of the cylinder before lifting, E2 = 0.5 * I2 * w^2 - kinetic energy of the pulley before lifting, E1' = 0 - potential energy of the cylinder after lifting (the center of mass of the cylinder remains at the same height ), E2' = 0.5 * I2 * w'^2 - kinetic energy of the pulley after lifting, where w' is the angular velocity of the pulley after lifting.

Taking into account the fact that at the initial moment of time the cylinder is at rest and the pulley rotates with an angular velocity ? = 20 rad/s, we get:

m * g * h = 0.5 * I2 * w^2,

where g is the acceleration of gravity, h is the height of the rise of the cylinder.

Thus, the speed of the cylinder will be equal to:

v = sqrt(2 * g * h)

This means that the modulus of momentum of the cylinder will be equal to:

p1 = m * sqrt(2 * g * h)

Substituting the known values, we get:

p1 = 50 * sqrt(2 * 9.81 * 1) ≈ 100

Answer: 100.

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In our digital goods store you can purchase the solution to problem 14.2.18 from the collection of problems O.?. Kepe. This digital product is an electronic pdf file containing a detailed solution to the problem with step-by-step instructions and detailed calculations.

To solve the problem, it is necessary to find the modulus of momentum of cylinder 1, which is lifted by a homogeneous cylinder 2 of radius R = 0.2 m at the angular speed of rotation of the pulley ? = 20 rad/s. The mass of the cylinder is m = 50 kg.

To solve the problem, you can use the law of conservation of momentum, which is formulated as follows: the sum of the quantities of motion of all bodies in a closed system remains unchanged. Thus, the modulus of momentum of cylinder 1 is equal to the modulus of momentum of the system of cylinder 1 and pulley 2 before the cylinder begins to rise. It is known that the quantities of motion of the pulley and the cylinder are equal in magnitude, that is: p1 = p2

For a pulley, the amount of motion is: p2 = I2 * w, where I2 is the moment of inertia of the pulley, w is its angular velocity. The moment of inertia of the pulley can be found using the formula: I2 = 0.5 * M2 * R^2, where M2 is the mass of the pulley, R is its radius. Thus, the amount of movement of the pulley will be: p2 = 0.5 * M2 * R^2 * w.

Similarly, for a cylinder the momentum can be written as: p1 = m * v, where v is the cylinder speed.

To find the speed of the cylinder, it is necessary to use the law of conservation of energy, which is formulated as follows: the total mechanical energy of a closed system remains unchanged. Thus, the total mechanical energy of the system before the cylinder begins to rise is equal to the total mechanical energy of the system after the cylinder rises: E1 + E2 = E1' + E2', where E1 = m * g * h - potential energy of the cylinder before lifting, E2 = 0.5 * I2 * w^2 - kinetic energy of the pulley before lifting, E1' = 0 - potential energy of the cylinder after lifting (the center of mass of the cylinder remains at the same height ), E2' = 0.5 * I2 * w'^2 - kinetic energy of the pulley after lifting, where w' is the angular velocity of the pulley after lifting.

Taking into account the fact that at the initial moment of time the cylinder is at rest and the pulley rotates with an angular velocity ? = 20 rad/s, we get: m * g * h = 0.5 * I2 * w^2, where g is the acceleration of gravity, h is the height of the cylinder, which in this case is equal to the radius of the pulley R.

Using the found value of the moment of inertia of the pulley and the angular velocity of rotation, you can find the module of the pulley momentum p2: p2 = 0.5 * M2 * R^2 * w = 0.5 * M2 * R^2 * ?.

Then, using the law of conservation of momentum, we can find the magnitude of the momentum of the cylinder p1: p1 = p2 = 0.5 * M2 * R^2 * ?.

Finally, using the found value of the modulus of momentum of the cylinder and the mass of the cylinder, you can find its speed v: v = p1 / m = (0.5 * M2 * R^2 * ?) / m.

Thus, we found the magnitude of the momentum of the cylinder and its speed using the laws of conservation of momentum and energy.

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Solution to problem 14.2.18 from the collection of Kepe O.?. consists in determining the modulus of momentum of cylinder 1, which lifts pulley 2 of radius R = 0.2 m at an angular velocity of rotation ? = 20 rad/s.

To solve the problem, it is necessary to use the law of conservation of momentum. Since the force of gravity acts on the cylinder, its momentum changes. However, since the system is closed, the change in the momentum of the cylinder must be compensated by a change in the momentum of the pulley.

The modulus of momentum of the cylinder can be calculated using the formula: p = mv, where m is the mass of the cylinder, v is its speed. Since the cylinder rises vertically, its speed is equal to the lifting speed, which can be expressed through the speed of rotation of the pulley: v = R?, where R is the radius of the pulley, ? - angular speed of rotation.

Thus, the modulus of momentum of the cylinder is p = mR?. Substituting the known values, we get: p = 50 kg * 0.2 m * 20 rad/s = 200 kg*m/s.

Answer: 100.

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