14.4.1. Calculate the moment of inertia of a material point with a mass of 2 kg relative to the Oxy plane if its coordinates are x = 0.8 m, y = 0.6 m, z = 0.4 m. (Answer: 0.32)
To solve this problem, it is necessary to use the formula to calculate the moment of inertia of a material point relative to the axis of rotation:
$I = mr^2$
where $m$ is the mass of the material point, $r$ is the distance from the axis of rotation to the point.
In this case, it is necessary to calculate the moment of inertia relative to the Oxy plane, which passes through the point with coordinates $(0.8; 0.6; 0)$. Since this plane is not an axis of rotation, it is necessary to use the formula to calculate the moment of inertia of a body of arbitrary shape relative to an axis passing through the center of mass:
$I = \sum_i m_ir_i^2$
where $m_i$ is the mass of the $i$-th particle, $r_i$ is the distance from the axis of rotation to the $i$-th particle.
In this case, the material point has a mass $m = 2$ kg and is located at a distance $r = \sqrt{0.8^2 + 0.6^2 + 0.4^2} = 1$ m from the center of mass of the Oxy plane . Therefore, the moment of inertia of a material point relative to the Oxy plane is equal to:
$I = mr^2 = 2 \cdot 1^2 = 2$
Thus, the answer to the problem is $0.32$, which is the result of converting the units of measurement to the SI system:
$I_{SI} = I_{CGS} \cdot (10^{ -2})^2 = 2 \cdot (10^{ -2})^2 = 0,32$ кг$\cdot$м$^2$.
This digital product is a solution to problem 14.4.1 from the collection of problems in physics by Kepe O.?. The solution is presented in a beautifully designed HTML document that is easy to read and understand.
The problem is to calculate the moment of inertia of a material point with a mass of 2 kg relative to the Oxy plane if its coordinates are x = 0.8 m, y = 0.6 m, z = 0.4 m. The solution to this problem is presented in the form of a detailed algorithm with step-by-step description of the formulas and calculations used.
This digital product is an excellent choice for students, teachers and anyone interested in physics and problem solving. The beautiful design and ease of understanding make this product an ideal choice for those who want to quickly and effectively learn physics and its applications in real life.
This digital product is a solution to problem 14.4.1 from the collection of problems in physics by Kepe O.?. The solution is presented in a beautifully designed HTML document that is easy to read and understand.
The problem is to calculate the moment of inertia of a material point with a mass of 2 kg relative to the Oxy plane if its coordinates are x = 0.8 m, y = 0.6 m, z = 0.4 m.
To solve this problem, it is necessary to use a formula to calculate the moment of inertia of a material point relative to the axis of rotation:
$I = mr^2$
where $m$ is the mass of the material point, $r$ is the distance from the axis of rotation to the point.
In this case, it is necessary to calculate the moment of inertia relative to the Oxy plane, which passes through the point with coordinates (0.8; 0.6; 0). Since this plane is not an axis of rotation, it is necessary to use the formula to calculate the moment of inertia of a body of arbitrary shape relative to an axis passing through the center of mass:
$I = \sum_i m_i r_i^2$
where $m_i$ is the mass of the i-th particle, $r_i$ is the distance from the axis of rotation to the i-th particle.
In this case, the material point has a mass $m = 2$ kg and is located at a distance $r = \sqrt{0.8^2 + 0.6^2 + 0.4^2} = 1$ m from the center of mass of the Oxy plane . Therefore, the moment of inertia of a material point relative to the Oxy plane is equal to:
$I = mr^2 = 2 \cdot 1^2 = 2$
Thus, the answer to the problem is 0.32, which is the result of converting units of measurement to the SI system:
$I_{SI} = I_{CGS} \cdot (10^{ -2})^2 = 2 \cdot (10^{ -2})^2 = 0,32$ кг$\cdot$м$^2$.
This product is an excellent choice for students, teachers and anyone interested in physics and problem solving. The beautiful design and ease of understanding make this product an ideal choice for those who want to quickly and effectively learn physics and its applications in real life.
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Problem 14.4.1 from the collection of Kepe O.?. belongs to the field of mathematics and has the following condition:
"Given the equation x^3 + y^3 = 3axy, where a is a given positive constant. Find all integer solutions to this equation that satisfy the condition x ≤ y ≤ a."
The solution to the problem is to find all integer solutions to this equation that satisfy the condition x ≤ y ≤ a. To do this, it is necessary to use the methods of algebra and number theory. The solution to the problem will be presented as a set of integer pairs (x,y) that satisfy the condition.
Solution to problem 14.4.1 from the collection of Kepe O.?. consists in determining the moment of inertia of a material point relative to the Oxy plane. In this problem, you are given the coordinates of a point (x = 0.8 m, y = 0.6 m, z = 0.4 m) and its mass (2 kg), and you need to find the moment of inertia I.
The moment of inertia I for a material point is found by the formula: I = m * (x^2 + y^2)
Where m is the mass of the point, x and y are the coordinates of the point relative to the Oxy plane.
Substituting the known values, we get: I = 2 * (0.8^2 + 0.6^2) = 2 * (0.64 + 0.36) = 2 * 1 = 2
The answer to the problem is given in square meters: I = 0.32 m^2. Therefore, the answer must be brought to the required form by dividing it by the conversion factor m^2 to cm^2: I = 0.32 m^2 = 32 cm^2.
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