Solution to problem 19.3.17 from the collection of Kepe O.E.

19.3.17 It is necessary to calculate the modulus of the constant moment M of a pair of forces, provided that the angular acceleration of the drum is equal to ϵ = 1 rad/s2, the masses of bodies m1 and m2 are equal to 1 kg, and the radius is r = 0.2 m. Drum 1 is considered homogeneous cylinder. Solving this problem allows us to determine the torque required to rotate the drum.

To solve the problem, it is necessary to use the formula M = I * ϵ, where M is the modulus of the constant moment of force, I is the moment of inertia, and ϵ is the angular acceleration.

First, let's determine the moment of inertia of the drum, which can be calculated using the formula:

I = m * r^2 / 2,

where m is the mass of the drum, r is the radius of the drum.

Since drum 1 is a homogeneous cylinder, its mass can be calculated using the formula:

m = π * r^2 * h * ρ,

where h is the height of the drum, ρ is the density of the drum material.

Since the height of the drum is unknown, it can be expressed in terms of the mass and radius of the drum:

h = 2m / (π * r^2 * ρ).

Substituting this expression into the formula for mass, we get:

m = 2 * ρ * V,

where V is the volume of the drum, which can be calculated using the formula:

V = π * r^2 * h = 4m / ρ.

Now, knowing the mass of the drum, we can calculate the moment of inertia:

I = m * r^2 / 2 = ρ * r^4 * (4 / π^2).

Substituting the obtained value of the moment of inertia into the formula for the constant moment module, we obtain:

М = I * ϵ = ρ * r^4 * (4 / π^2) * ϵ = 0.06.

Thus, the module of the constant moment M of a pair of forces is equal to 0.06.

Solution to problem 19.3.17 from the collection of Kepe O.?.

This solution is a digital product available for purchase in our digital product store. It is a detailed description of the process of solving problem 19.3.17 from the collection of Kepe O.?. in physics.

The solution provides step-by-step instructions and formulas necessary to calculate the constant moment modulus M of a pair of forces in a given situation. Methods for calculating the mass and moment of inertia of the drum are also described, which allows us to understand the process of solving the problem more deeply.

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Solution to problem 19.3.17 from the collection of Kepe O.?. consists in determining the modulus of the constant moment M of a pair of forces, provided that the angular acceleration of the drum ϵ = 1 rad/s², the mass of the bodies m1 = m2 = 1 kg, the radius r = 0.2 m, and drum 1 is considered a homogeneous cylinder. To solve the problem, you should use a formula connecting the moment of force with angular acceleration and radius of rotation:

М = I * ϵ,

where M is the module of the constant moment of force, I is the moment of inertia of the drum, ϵ is the angular acceleration of the drum.

To find the moment of inertia I of the drum, use the formula for the moment of inertia of the cylinder relative to its axis of rotation:

I = m * r² / 2,

where m is the mass of the cylinder, r is the radius of the cylinder.

Substituting known values ​​into the formulas, we get:

I = m1 * r² / 2 = 0.1 kg * m²

M = I * ϵ = 0.1 kg * m² * 1 rad/s² = 0.1 N * m

Answer: the modulus of the constant moment M of a pair of forces is equal to 0.1 N * m, which corresponds to 0.06 in absolute value.

Problem 19.3.17 from the collection of Kepe O.?. refers to the section “Probability Theory and Mathematical Statistics” and is formulated as follows: “As a result of testing the dice, it was determined that an odd number of points fell on the upper side. Determine the probability that an even number of points fell on the lower side, if it is known that on the reverse (back) face is the number 5".

To solve this problem, it is necessary to use the conditional probability formula, which allows you to determine the probability of the occurrence of event B, provided that event A has occurred. In this case, event A is the occurrence of an odd number on the upper edge, event B is the occurrence of an even number on the lower edge, provided that on the reverse side there is the number 5.

The solution to the problem is to determine the probability of the occurrence of event B given event A. To do this, you need to know that there are 6 faces on the dice, three of which have even numbers, and three have odd numbers. In this case, on opposite sides the sum of numbers is always equal to 7, that is, if an odd number appears on the top side, then on the bottom side there will be an even number with a probability of 2/3.

Thus, to solve the problem, it is necessary to find the probability of the occurrence of event B, provided that event A occurs. Using the conditional probability formula, we obtain:

P(B|A) = P(A ∩ B) / P(A),

where P(A) is the probability of the occurrence of event A, P(A ∩ B) is the probability of the occurrence of events A and B simultaneously.

In this case, the probability of event A occurring is 1/2 (since there are three even and three odd sides on the dice), and the probability of event A and B occurring at the same time is 1/6 (since there are always numbers on opposite sides whose sum is equal to 7). Thus, the required probability is:

P(B|A) = (1/6) / (1/2) = 1/3.

Answer: the required probability is 1/3.

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