Total moment of inertia of a two particle system

In summary: So in this case, it would be K_total = Ka + Kb = 0.5m(3*omega*r)^2 + 0.5m(omega*r)^2 = 1.5m(omega*r)^2.
  • #1
Linus Pauling
190
0
1. Find the moment of inertia I_x of particle a with respect to the x-axis (that is, if the x-axis is the axis of rotation), the moment of inertia I_y of particle a with respect to the y axis, and the moment of inertia I_z of particle a with respect to the z axis (the axis that passes through the origin perpendicular to both the x and y axes).

Particle a is located 3r from the y axis, and particle b is located r away.




2. I = mr^2
I = SUMM(m_i*r_i^2)




3. I_a = 9mr^2
I_b = mr^2

Total I = 10mr^2

No idea what I am doing wrong
 
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  • #2
Linus Pauling said:
Particle a is located 3r from the y axis, and particle b is located r away.
Give the complete coordinates and the mass of each particle.
 
  • #3
Particle a:

distance from y-axis = 3r
distance from x-axis = r

Particle b:

distance from y-axis = r
distance from x-axis = -4r

It doesn't say anything explicitly about mass, and so I think we assume each particle has mass m.
 
  • #4
Linus Pauling said:
3. I_a = 9mr^2
I_b = mr^2

Total I = 10mr^2

No idea what I am doing wrong
Looks like you've found I_y. What about I_x?
 
  • #5
I_x would be m(r)^2 + m(-4r)^2 = mr^2 + 16mr^2 = 17mr^2.

Am I simply adding that to I_y to get I_total? It asks for total I with respect to the y axis, though, why would I need I_x?
 
  • #6
Shoot, just realized I posted to wrong question. Here's what I am trying to answer:

Find the total moment of inertia I of the system of two particles shown in the diagram with respect to the y axis
 
  • #7
Linus Pauling said:
Here's what I am trying to answer:

Find the total moment of inertia I of the system of two particles shown in the diagram with respect to the y axis
Assuming the diagram matches your description, your answer seems correct to me. (Post the diagram, if you can.)
 
  • #8
134726A.jpg
 
  • #9
Looks good to me. I say your answer is correct.
 
  • #10
Ok, I requested the correct answer: 11mr^2

WTF?!
 
  • #11
Linus Pauling said:
Ok, I requested the correct answer: 11mr^2
Sounds bogus to me. You might want to mention this to your instructor.
 
  • #12
Ok, now:

Using the formula for kinetic energy of a moving particle K=\frac{_1}{^2}mv^2, find the kinetic energy K_a of particle a and the kinetic energy K_b of particle b.
Express your answers in terms of m, omega, and r separated by a comma.

I know I need to calculate linear speed, how do I do that? I know it's v=omega*r but I don't see how to put this together...
 
  • #13
Linus Pauling said:
I know I need to calculate linear speed, how do I do that?
What's the relationship between tangential speed and angular speed for something going in a circle?
 
  • #14
Just edited before you replied: v=omega*r

But what are omega and r here?!
 
  • #15
Particle A:

Ka = 0.5m(3*omega*r)^2

Kb = 0.5m(omega*r)^2

Apparently Kb is wrong but I did it the same way as Ka....
 
  • #16
Ok I got Kb = m(omega*r)^2

What is total kinteic eneregy?
 
  • #17
Linus Pauling said:
Ka = 0.5m(3*omega*r)^2

Kb = 0.5m(omega*r)^2

Apparently Kb is wrong but I did it the same way as Ka....
These both look OK to me.

Linus Pauling said:
Ok I got Kb = m(omega*r)^2
How did you get that? (Are you posting this problem exactly as given, word for word?)

What is total kinetic energy?
Just add up the kinetic energy of each mass.
 

What is the Moment of Inertia?

The moment of inertia, often denoted as I, is a measure of an object's resistance to rotational motion around a particular axis. It depends on both the mass and the distribution of mass relative to the axis of rotation. In simpler terms, it tells us how difficult it is to change the rotational motion of an object.

How is the Total Moment of Inertia of a Two-Particle System Calculated?

The total moment of inertia (\(I_{\text{total}}\)) of a two-particle system is the sum of the individual moments of inertia (\(I_1\) and \(I_2\)) of the two particles with respect to the same axis. Mathematically, it is expressed as:

\[I_{\text{total}} = I_1 + I_2\]

What is the Formula for the Moment of Inertia of a Particle?

The moment of inertia (\(I\)) of a particle with mass \(m\) and distance \(r\) from the axis of rotation is calculated using the formula:

\[I = m \cdot r^2\]

How Do You Calculate the Moment of Inertia for Each Particle in a Two-Particle System?

To calculate the moment of inertia for each particle in a two-particle system, you need to know the mass of each particle and its distance from the axis of rotation. For each particle:

\[I_i = m_i \cdot r_i^2\]

Where \(I_i\) is the moment of inertia of the \(i\)-th particle, \(m_i\) is its mass, and \(r_i\) is its distance from the axis of rotation.

What Are the Units of Moment of Inertia?

The units of moment of inertia depend on the units of mass and distance used in the calculation. In the International System of Units (SI), the units of moment of inertia are typically kilogram-square meters (kg·m²).

Why is the Total Moment of Inertia Important?

The total moment of inertia of a system of particles is important in physics and engineering because it determines how the system responds to rotational forces and accelerations. It plays a crucial role in problems involving rotational motion, such as calculating the angular acceleration of an object subjected to a torque or understanding the behavior of rotating systems like wheels, disks, or celestial bodies.

What Happens to the Total Moment of Inertia if Particles are Moved Closer or Farther from the Axis of Rotation?

The total moment of inertia of a two-particle system will change if the particles are moved closer or farther from the axis of rotation. Increasing the distance of a particle from the axis of rotation (increasing \(r_i\)) will generally increase its contribution to the total moment of inertia, making the system more resistant to changes in rotational motion. Conversely, moving a particle closer to the axis will decrease its contribution to the total moment of inertia.

Can the Total Moment of Inertia of a System be Negative?

No, the total moment of inertia of a system cannot be negative. Moment of inertia is always a positive or zero value, representing the rotational resistance of the system. Negative values do not have physical significance in the context of moment of inertia.

Is the Total Moment of Inertia Affected by the Orientation of the Axis of Rotation?

Yes, the total moment of inertia can be affected by the orientation of the axis of rotation. In a three-dimensional space, the moment of inertia depends on the choice of the axis. Different orientations of the axis can result in different moment of inertia values for the same system of particles.

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