Solution to problem 15.6.4 from the collection of Kepe O.E.

15.6.4 A rotor with a mass m = 314 kg and a radius of gyration relative to the axis of rotation equal to 1 m is given an angular velocity ?0 = 10 rad/s. Left to its own devices, it stopped after making 100 revolutions. Determine the friction moment in the bearings, considering it constant. (Answer 25)

Given a rotor with a mass of 314 kg and a radius of gyration of 1 m, rotating at a speed of 10 rad/s. After being left to its own devices, it stopped after 100 revolutions. It is necessary to determine the friction moment in the bearings, assuming it is constant. The answer to the problem is 25.

Solution to problem 15.6.4 from the collection of Kepe O.?.

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The digital product is a solution to problem 15.6.4 from the collection of Kepe O.?. in physics. The problem considers a rotor with a mass of 314 kg and a radius of gyration of 1 m, which rotates at a speed of 10 rad/s. After leaving the rotor to its own devices, it stopped after 100 revolutions. It is necessary to determine the friction moment in the bearings, assuming it is constant. The answer to the problem is 25.

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The product, the description of which is required, is not a physical object, but is a problem from a collection of problems in physics by Kepe O.?.

Problem 15.6.4 states:

“A rotor with a mass m = 314 kg and a radius of gyration relative to the axis of rotation equal to 1 m is given an angular velocity ?0 = 10 rad/s. Left to its own devices, it stopped after making 100 revolutions. Determine the friction moment in the bearings, considering it constant. (Answer 25)"

From the problem it is known that a rotor with a mass of 314 kg and a radius of gyration of 1 m had an initial angular velocity of 10 rad/s, and then it stopped after 100 revolutions. It is required to find the friction moment in the bearings, considering it constant.

The solution to this problem can be found using the laws of conservation of energy and angular momentum. After 100 revolutions, the rotor stopped, having lost all the kinetic energy that was on it at the initial moment of time. Consequently, the moment of friction forces in the bearings acting on the rotor must be equal to the moment of impulse of the rotor at the initial moment of time.

The rotor angular momentum can be calculated using the formula:

L = I * w,

where L is the moment of impulse, I is the moment of inertia of the rotor, w is the angular velocity.

In this case, the moment of inertia I = m * r^2 = 314 * 1^2 = 314 kg * m^2, where r is the radius of the rotor.

Thus, L = 314 * 10 = 3140 kg * m^2/s.

From the law of conservation of angular momentum it follows that the frictional moment in the bearings must be equal to the rotor angular momentum at the initial moment of time:

M = L / t,

where t is the time during which the rotor stopped.

Since the rotor made 100 revolutions, it traveled the path:

S = 2 * pi * r * n = 2 * 3.14 * 1 * 100 = 628 m.

Since the angular velocity of the rotor is constant, the time during which the rotor stopped can be calculated using the formula:

t = w0 / a,

where a is the angular acceleration equal to -w0^2 / 2 * pi * n.

w0 is the initial angular velocity.

Then:

t = w0 / (-w0^2 / 2 * pi * n) = -2 * pi * n / w0 = -2 * 3.14 * 100 / 10 = -62.8 с.

Since time cannot be negative, we should take the time module: t = 62.8 s.

Thus, the friction moment in the bearings can be calculated:

M = L / t = 3140 / 62.8 = 50 Н * м.

Answer: 50 N*m.

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