Moment of inertia of a three particle system

In summary, the conversation is about finding the moment of inertia for a system of three particles located at (-a,-a), (a,-a), and (0,a) for an axis along the z-axis. The equations for center of mass and moment of inertia are provided and the attempt at a solution is shown. However, the calculated answer does not match the answer given in the book and the person asking for help is unsure where they made a mistake.
  • #1
Avatrin
245
6

Homework Statement


I have three particles of mass m at (-a,-a), (a,-a) and (0,a)

Find the moment of inertia I_z around the center of mass of the system for an axis along the z-axis.

Homework Equations


Center of mass:
CM = Ʃmr

Moment of inertia:
I = Ʃmp^2

m = mass
r = position
p = distance from axis

The Attempt at a Solution


I found that the center of mass is CM = (0,-a/3);
CM_x = (-am + am + 0m)/3m = 0
CM_y = (-a-a+a)m/3m = -a/3
This is something I know is correct..

The real problem starts with the moment of inertia:
p_1 = 4a/3
p_2 = p_3 = √(a^2 + (2a/3)^2)

(p_1)^2 = (16/9)a^2
(p_2)^2 = (a^2 + (2a/3)^2) = (13/9)a^2

Ʃmp^2 = m(16/9)a^2 + m(13/9)a^2 + m(13/9)a^2
= m((16/9)a^2 + (26/9)a^2) ≈ 4.67ma^2

The problem is that the book says that the solution is 4.01ma^2. Where have I made a mistake?
 
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  • #2
Avatrin said:
The problem is that the book says that the solution is 4.01ma^2. Where have I made a mistake?

I got the answer that you had as well. Make sure that you've read the question correctly.
 
  • #3
Last edited:

1. What is the definition of moment of inertia?

The moment of inertia of a three particle system is a measure of its resistance to rotational motion. It is defined as the sum of the mass of each particle multiplied by the square of its distance from the axis of rotation.

2. How is the moment of inertia calculated for a three particle system?

The moment of inertia for a three particle system can be calculated by using the formula I = m1r1² + m2r2² + m3r3², where m1, m2, and m3 are the masses of the particles and r1, r2, and r3 are the distances of the particles from the axis of rotation.

3. What factors affect the moment of inertia of a three particle system?

The moment of inertia of a three particle system is influenced by the mass and distribution of the particles. A larger mass or a farther distance from the axis of rotation will result in a higher moment of inertia.

4. How does the moment of inertia of a three particle system relate to its rotational motion?

The moment of inertia is directly proportional to the angular acceleration of the three particle system. A higher moment of inertia will result in a slower rotational motion, while a lower moment of inertia will result in a faster rotational motion.

5. Can the moment of inertia of a three particle system be greater than the sum of its individual particles?

Yes, the moment of inertia of a three particle system can be greater than the sum of its individual particles. This is because the moment of inertia takes into account the distribution of mass, not just the total mass of the particles.

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