Moments of Inertia of two particles

In summary, the problem involves two particles attached to a diameter of a uniform circular disk and free to rotate about a horizontal axis through a point on the diameter. The task is to determine the moment of inertia of the system in terms of integer constants a, b, and m. One approach is to find the moment of inertia of the entire system about its center of mass and use the parallel axis theorem. Another approach is to modify the generic formula for the moment of inertia of a circular disk about its center, taking into account the specific mass and axis of rotation in this problem.
  • #1
Spaceflea
2
0

Homework Statement



Two particles, each of mass m, are attached one to each end of a diameter PQ of a uniform circular disk, of mass 4m, radius a with its centre at O. The system is free to rotate about a horizontal axis through A, a point on PQ such that OA = b as indicated in the diagram below. The system is released from rest when PQ is horizontal.[/B]
Determine the Moment of Inertia of the system about the axis A, in terms of integer constants, a , b and m.

Homework Equations


I=ma^2
parallel axis equation I=Icentre of mass + md^2

The Attempt at a Solution


I have determined the moment of inertia of the disc to be 0.5ma^2.
I have determined the moment of inertia of Q to be m(b+a)^2.
I have determined the moment of inertia of P to be m(a-b)^2.
Therefore I should just add these three numbers together to make a total moment of inertia, but this answer is marked wrong (it is marked by a computer programme). Where am I going wrong in my method? Thanks.
 
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  • #2
Spaceflea said:
I have determined the moment of inertia of the disc to be 0.5ma^2.
That's the generic formula for a circular disk about its center. You'll have to modify it.

Spaceflea said:
I have determined the moment of inertia of Q to be m(b+a)^2.
I have determined the moment of inertia of P to be m(a-b)^2.
These are OK.

Spaceflea said:
Therefore I should just add these three numbers together to make a total moment of inertia, but this answer is marked wrong (it is marked by a computer programme). Where am I going wrong in my method?
Your expression for the moment of inertia of the disk is incorrect: You have the wrong mass and the wrong axis.

Correct that, and your method should work fine.

Another approach would be to find the moment of inertia of the entire system about its center of mass, then use the parallel axis theorem. Do it both ways and compare!
 
  • #3
Thanks so much for the reply! But I am confused over which axis and mass I should use instead. Obviously not point A...
 
  • #4
Spaceflea said:
But I am confused over which axis and mass I should use instead. Obviously not point A...
I assume you are talking about the disk? What's the mass of the disk? You need its moment of inertia about point A, but don't start there. Start with the moment of inertia about the center of mass and use the parallel axis theorem.
 
  • #5


Your approach is correct, but there may be a mistake in your calculation of the moment of inertia for particle Q. It should be m(b+a)^2, not m(b+a)^3. Double check your calculation for Q and see if that resolves the issue. Also, make sure you are using the correct formula for the moment of inertia of a point particle, which is I=mr^2, where r is the distance from the particle to the axis of rotation.
 

1. What is the definition of the moment of inertia of two particles?

The moment of inertia of two particles is a measure of an object's resistance to rotational motion. It is calculated by multiplying the mass of each particle by its distance from the axis of rotation squared, and then adding these values together.

2. How is the moment of inertia of two particles different from the moment of inertia of one particle?

The moment of inertia of two particles is different from the moment of inertia of one particle because it accounts for the distribution of mass in the object. In the case of two particles, the distance between the particles also affects the moment of inertia.

3. What is the formula for calculating the moment of inertia of two particles?

The formula for calculating the moment of inertia of two particles is I = m1r1^2 + m2r2^2, where I is the moment of inertia, m1 and m2 are the masses of the two particles, and r1 and r2 are the distances of the particles from the axis of rotation.

4. How does the moment of inertia of two particles affect the rotational motion of an object?

The moment of inertia of two particles affects the rotational motion of an object by determining how easily it can rotate. Objects with higher moments of inertia will have a greater resistance to rotational motion, and will require more torque to rotate.

5. What are some real-life applications of understanding the moment of inertia of two particles?

Understanding the moment of inertia of two particles is important in various fields such as engineering, physics, and sports. It is used in designing structures and machines that need to rotate, as well as in predicting the motion of objects in sports such as figure skating and gymnastics.

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