Solution to problem 15.7.7 from the collection of Kepe O.E.

15.7.7. Suppose that we have two homogeneous cylindrical rollers weighing 20 kg each, designated as roller 1 and roller 2. If we set them in motion from rest by a constant moment of a pair of forces M = 2 N • m, then what will be the speed of the roller axes, when will they move a distance of 3 meters? The radii of the rollers are equal to R1 = R2 = 0.2 meters. The answer to this problem is 1.

Solution to problem 15.7.7 from the collection of Kepe O.?.

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Problem 15.7.7 is to determine the speed of the axes of two homogeneous cylindrical rollers weighing 20 kg each when they are driven by a constant moment of a pair of forces M = 2 N • m from rest to a distance of 3 meters. The radii of the rollers are equal to R1 = R2 = 0.2 meters.

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Solution to problem 15.7.7 from the collection of Kepe O.?. consists in determining the speed of the axes of two homogeneous cylindrical rollers, each weighing 20 kg, when they move over a distance of 3 m. The rollers are driven from rest by a constant moment of a pair of forces M = 2 N • m. The radii of the rollers are R1 = R2 = 0, 2 m.

To solve the problem, it is necessary to use the laws of the dynamics of rotational motion. Using the angular momentum formula L = Iω, where I is the moment of inertia of the body, and ω is its angular velocity, we can determine the angular velocity of rotation of the rollers at a given moment of force.

It is known that the moment of inertia of a cylinder relative to its axis passing through the center of mass is equal to I = (1/2)mr^2, where m is the mass of the body and r is the radius. Substituting the values, we get I = (1/2)20 kg*(0.2 m)^2 = 0.4 kg*m^2.

Next, using the formula for the moment of force M = dL/dt, you can determine the angular acceleration α, and then the angular velocity ω. After this, knowing the radius of the rollers and the angular velocity, you can determine the linear speed of their axes.

Substituting the values, we get:

M = Iα α = M/I = 2 N•m / 0.4 kg•m^2 = 5 rad/s^2

ω = αt = 5 rad/s^2 * 3 s = 15 rad/s

v = ωR = 15 rad/s * 0.2 m = 3 m/s

Thus, the speed of the roller axes when they move over a distance of 3 m is 3 m/s. Answer: 1.

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