Moment of Inertia of four particles

In summary, the problem involves finding the moment of inertia of a four-particle system with masses of 3.0 kg and 4.0 kg, connected at the corners of a square. The moment of inertia is found using the parallel-axis theorem and the result for Problem 41. A check is also done by direct computation, resulting in two different answers. The final moment of inertia is found to be 9.33 kgm^2.
  • #1
Rubidium
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Homework Statement


Four particles, one at each of the four corners of a square with 2.0-m-long edges, are connected by massless rods. The masses are m1=m3=3.0 kg and m2=m4=4.0 kg. Find the moment of inertia of the system about the z axis. (the z axis runs through m2, which is at the origin, m1 is on the y axis, and m3 is on the x axis.
Use the parallel-axis theorem and the result for Problem 41 to find them moment of inertia of the four-particle system about an axis that passes through the center of mass and is parallel with the z axis. Check your result by direct computation.

Homework Equations


I don't know what they mean by direct computation but I get two different answers.

The Attempt at a Solution


I=m1r1^2+m2r2^2+m3r3^2+m4r4^2=(3kg)(2m)^2+(4kg)(0)+(3kg)(2m)^2+(4kg)(2sqrt(2))^2=56kgm^2
Icm=I-Mh^2=56-(14kg)(4)=28kgm^2
Then integrating for Icm I get 9.33 as a result.
 
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  • #2
Since the four mass are at equal distances from the axis passing through cm, you can find Icm by using the formula sigma(m)*r^2
 
  • #3


I would respond by saying that the moment of inertia (I) is a measure of an object's resistance to rotational motion and is dependent on both the mass and distribution of the particles in the system. In this case, the moment of inertia is being calculated about the z-axis, which is perpendicular to the plane of the square and passes through the center of mass.

To find the moment of inertia of the system about the z-axis, we can use the parallel-axis theorem, which states that the moment of inertia about an axis parallel to an axis passing through the center of mass is equal to the moment of inertia about the center of mass plus the product of the mass and the square of the distance between the two axes.

Using this theorem, we can calculate the moment of inertia (Iz) about the z-axis as:

Iz = Icm + Md^2

Where Icm is the moment of inertia about the center of mass, M is the total mass of the system, and d is the distance between the z-axis and the center of mass.

Substituting the values given in the problem, we get:

Iz = 28kgm^2 + (7kg)(2sqrt(2)m)^2 = 56kgm^2

This result is the same as the direct computation method used in the attempt at a solution, which is a good check for the accuracy of the calculation.

In conclusion, the moment of inertia of the four-particle system about the z-axis is 56kgm^2. This value represents the resistance to rotational motion of the system and can be used in further calculations involving the rotational dynamics of the system.
 

What is moment of inertia?

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It is calculated by summing the mass of each particle in the object multiplied by the square of its distance from the axis of rotation.

How is moment of inertia calculated for four particles?

The moment of inertia for four particles is calculated by adding the individual moments of inertia for each particle. This can be done by using the parallel axis theorem, which states that the moment of inertia of a group of particles is equal to the moment of inertia of the group's center of mass plus the sum of the particles' masses multiplied by the square of their distance from the center of mass.

What factors affect the moment of inertia of four particles?

The moment of inertia of four particles is affected by the mass and distribution of the particles. Objects with more mass or particles located farther from the axis of rotation will have a higher moment of inertia.

How does moment of inertia relate to an object's shape?

The moment of inertia is affected by an object's shape. Objects with a larger radius of gyration (the distance from the axis of rotation to a point where all of the object's mass could be concentrated) will have a higher moment of inertia.

Why is moment of inertia important in physics?

Moment of inertia plays a crucial role in determining an object's rotational motion. It is used in equations to calculate an object's angular acceleration and is an important factor in understanding how objects move and behave in rotational motion.

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