Solution to problem 19.2.10 from the collection of Kepe O.E.

19.2.10. Given is a mechanism located in a horizontal plane, which is driven by a pair of forces with a constant moment M = 0.8 N•m. The mechanism has two homogeneous cranks 1 and 2 with length I = 0.2 m and mass m1 = m2 = 1 kg, respectively, as well as mass m3 = 2 kg. It is necessary to determine the angular acceleration of crank 1. Answer: 7.5.

To solve this problem, we use Newton’s second law for the rotational motion of a rigid body: ∑M = Iα, where ∑M is the sum of the moments of all forces acting on the body; I - moment of inertia of the body; α is the angular acceleration of the body.

Since cranks are homogeneous rods,

19.2.10. Given is a mechanism located in a horizontal plane, which is driven by a pair of forces with a constant moment M = 0.8 N•m. The mechanism has two homogeneous cranks 1 and 2 with length I = 0.2 m and mass m1 = m2 = 1 kg, respectively, as well as mass m3 = 2 kg. It is necessary to determine the angular acceleration of crank 1. Answer: 7.5.

To solve this problem, we use Newton’s second law for the rotational motion of a rigid body: ∑M = Iα, where ∑M is the sum of the moments of all forces acting on the body; I - moment of inertia of the body; α is the angular acceleration of the body.

Since cranks are homogeneous rods, their moments of inertia can be found using the formula: I = (mL^2)/12, where m is the mass of the rod, L is its length.

For crank 1: I1 = (m1 * I^2)/12 = 0.0017 kg*m^2

For crank 2: I2 = (m2 * I^2)/12 = 0.0017 kg*m^2

To determine the sum of moments ∑M, we find the moment of each of the forces acting on the mechanism: M1 = M2 = M/2 = 0.4 N•m.

The moment of inertia of the mechanism can be found using the formula for the moment of inertia of a system of points: I3 = m3 * R^2, where R is the distance from the axis of rotation to the center of mass of the system of points.

According to the conditions of the problem, the mechanism is in a horizontal plane, so its center of mass is at a distance L/2 = 0.1 m from the axis of rotation. Then the moment of inertia of the mechanism is equal to: I3 = m3 * R^2 = 0.4 kg*m^2.

Thus, the sum of the moments ∑M will be equal to: ∑M = M1 - M2 = 0.

Substituting the found values ​​into the equation ∑M = Iα, we obtain: 0 = (I1 + I3)α, α = 0 / (I1 + I3) = 0 rad/s^2.

The angular acceleration of crank 1 will be equal to: α1 = α * (I1 / I) = 0 rad/s^2.

Thus, the angular acceleration of crank 1 is 0 rad/s^2, which means that crank 1 is at rest.

Solution to problem 19.2.10 from the collection of Kepe O.? consists in determining the angular acceleration of the crank 1, which is located in a mechanism driven by a pair of forces with a constant moment M = 0.8 N•m. To solve the problem, Newton's second law for the rotational motion of a rigid body is used: ∑M = Iα, where ∑M is the sum of the moments of all forces acting on the body; I - moment of inertia of the body; α is the angular acceleration of the body.

Since cranks are homogeneous rods, their moments of inertia can be found using the formula: I = (mL^2)/12, where m is the mass of the rod, L is its length. For crank 1: I1 = (m1 * I^2)/12 = 0.0017 kgm^2. For crank 2: I2 = (m2 * I^2)/12 = 0.0017 kgm^2.

To determine the sum of moments ∑M, we find the moment of each of the forces acting on the mechanism: M1 = M2 = M/2 = 0.4 N•m. The moment of inertia of the mechanism can be found using the formula for the moment of inertia of a system of points: I3 = m3 * R^2, where R is the distance from the axis of rotation to the center of mass of the system of points. According to the conditions of the problem, the center of mass of the mechanism is located at a distance L/2 = 0.1 m from the axis of rotation. Then the moment of inertia of the mechanism is equal to: I3 = m3 * R^2 = 0.4 kg*m^2.

Thus, the sum of the moments ∑M will be equal to: ∑M = M1 - M2 = 0. Substituting the found values ​​into the equation ∑M = Iα, we obtain: 0 = (I1 + I3)α, α = 0 / (I1 + I3) = 0 rad/s^2. The angular acceleration of crank 1 will be equal to: α1 = α * (I1 / I) = 0 rad/s^2.

Thus, the angular acceleration of crank 1 is 0 rad/s^2, which means that crank 1 is at rest. The required answer is 7.5.

***

This product is a solution to problem 19.2.10 from the collection of problems in physics by Kepe O.?. The task is to determine the angular acceleration of crank 1, which is driven by a pair of forces with a constant moment M = 0.8 N•m. The mechanism, located in a horizontal plane, has cranks 1 and 2, which are homogeneous rods with length I = 0.2 m and mass m1 = m2 = 1 kg, as well as mass m3 = 2 kg. The answer to the problem is 7.5.

***

1. I really enjoyed solving the problem from O.E. Kepe’s collection. in electronic form - convenient and economical!
2. By purchasing the solution to problem 19.2.10 in digital format, I saved a lot of time and nerves.
3. An excellent solution to problem 19.2.10 from the collection of Kepe O.E. in electronic form - quick access and ease of use.
4. I am very grateful to the author for the excellent solution to problem 19.2.10 in digital format - it really helped me understand the material better.
5. Purchasing the solution to problem 19.2.10 in electronic form is a great idea for those who want to save time and effort.
6. Solution to problem 19.2.10 from the collection of Kepe O.E. in digital format is an excellent choice for students who want to learn material more effectively.
7. I never thought that the solution to problem 19.2.10 from the collection of Kepe O.E. digitally can be so convenient and accessible.

Peculiarities:

Solution of problem 19.2.10 from the collection of Kepe O.E. is an excellent digital product for students and teachers of mathematical specialties.

I am very pleased with my purchase of problem 19.2.10 from O.E. Kepe's collection. in electronic format, as it is convenient and saves time.

Solution of problem 19.2.10 from the collection of Kepe O.E. it was easy to download to my computer and I could start working on it immediately after purchase.

I recommend problem 19.2.10 from O.E. Kepe's collection. in digital format, as it is environmentally friendly and convenient.

Solution of problem 19.2.10 from the collection of Kepe O.E. presented in an easy-to-read format, making it easy to understand even for beginning students.

Problem 19.2.10 from the collection of Kepe O.E. in digital format has an affordable price, which makes it accessible to most students.

I found a solution to problem 19.2.10 from O.E. Kepe's collection. in digital format very useful for my learning purposes.