Rotational Inertia: Calc. & Guidance Needed

In summary, the individual disks have a MoI about their own center of 0.000000011 kg * m^2 and a MoI about O of 0.00002475 kg * m^2.
  • #1
62
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Homework Statement
The figure shows an arrangement of 15 identical disks that have been glued together in a rod-like shape of length L = 1.6900 m and (total) mass M = 104.0 mg. The arrangement can rotate about a perpendicular axis through its central disk at point O. (a) What is the rotational inertia of the arrangement about that axis? Give your answer to four significant figures. (b) If we approximated the arrangement as being a uniform rod of mass M and length L, what percentage error would we make in using the formula in Table 10-2e to calculate the rotational inertia?
Relevant Equations
I = 1/12 ML^2
I = 1/2 mr^2
This question is ridiculously complex for me. I know how to find the Inertia if it was a rod

##I =\frac{1}{12} ML^{2} = \frac{1}{12}(0.000104)(1.6900)^2 =0.00002475 kg * m^2##

To find the Inertia for the individual disks I am having trouble. Not sure how to add them all up together. Guidance would be greatly appreciated.
 

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  • #2
Riman643 said:
To find the Inertia for the individual disks I am having trouble
Consider the nth disk from O, to the right, say.
What is its MoI about its own centre?
How far is its centre from O?
Using the parallel axis theorem, what is its MoI about O?
 
  • #3
I think I get where you are going. Every disk will be:

##I = \frac{1}{2} mr^{2} = \frac{1}{2} (\frac{0.000104}{15})(\frac{1.6900}{\frac{15}{2}})^{2} = 0.000000011 kg * m^2##

Then using the parallel axis theorem it would be:
##I = \frac{1}{2}mr^{2} + md^{2}##

but which disk do I use for the ##md^{2}## part? And would the distance to let's say the one to the right and left of the center bed twice the radius?
 
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  • #4
Riman643 said:
Every disk will be
How many radii make up length L?
Riman643 said:
would the distance to let's say the one to the right and left of the center bed twice the radius?
Yes. What about the next ones along?
 
  • #5
haruspex said:
How many radii make up length L?

15? Would that then make the parallel axis theorem in this particular case:

##I_{O} = 15(\frac{1}{2}mr^{2}) + md^{2}##
haruspex said:
Yes. What about the next ones along?

The next ones along would be 4 times the length of the radius. So I should use the furthest one away? 14 times the length of the radius?
 
  • #6
Riman643 said:
15?
No, try again.
Riman643 said:
The next ones along would be 4 times the length of the radius.
Yes.
Riman643 said:
So I should use the furthest one away?
No, each makes the contribution it makes. Sum the series.
 
  • #7
haruspex said:
No, try again.
Oops. It's 30 that will make up the entire length

haruspex said:
No, each makes the contribution it makes. Sum the series.
So would a correct formula with a series of distances be:

##I_{O} = 15mr^{2} + 2mr^{2}(2 + 4 + 6 + 8 +10 +12 + 14)##
 
  • #8
Riman643 said:
Oops. It's 30 that will make up the entire lengthSo would a correct formula with a series of distances be:

##I_{O} = 15mr^{2} + 2mr^{2}(2 + 4 + 6 + 8 +10 +12 + 14)##
Close...
There's something you left out on the 2, 4, 6 etc.
 
  • #9
haruspex said:
Close...
There's something you left out on the 2, 4, 6 etc.
Oops. took out the square when I took out ##r^{2}##. I got the answer right for the first part. But for some reason I am struggling getting the percentage with the rod wrong. For the rod I get ##24.75 mg * m^{2}## and for the actual Inertia I get ##25.02 mg * m^{2}##. Then I take ##(1 - \frac{24.75}{25.02}) * 100 = 1.067 %##. Not sure what I am doing wrong with that.
 

1. What is rotational inertia?

Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to changes in its rotational motion. It depends on the mass, shape, and distribution of mass of the object.

2. How is rotational inertia calculated?

The rotational inertia of an object can be calculated by multiplying the mass of the object by the square of its distance from the axis of rotation. It can also be calculated using the formula I = ∫r² dm, where r is the distance from the axis of rotation and dm is the infinitesimal mass element.

3. What are some real-life examples of rotational inertia?

Some common examples of rotational inertia include a spinning top, a spinning bicycle wheel, and a figure skater spinning on ice. In each of these examples, the object's rotational inertia is responsible for keeping it in motion.

4. How does rotational inertia affect the motion of objects?

Rotational inertia plays a crucial role in determining how objects rotate and how their motion changes. Objects with a higher rotational inertia will resist changes in their rotational motion and will require more force to accelerate or decelerate.

5. How can understanding rotational inertia be useful?

Understanding rotational inertia is essential for engineers and scientists in designing and analyzing the motion of objects, such as in the construction of vehicles, machines, and structures. It also has practical applications in sports, such as figure skating and gymnastics, where rotational inertia is utilized to perform complex maneuvers.

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