Moment of inertia and angular KE confusion

In summary, the moment of inertia for a ring or circle is MR^2, where M is the total mass and R is the radius. There is no number prefix in front of the formula because all points on the ring or circle are at the same distance from the center. To represent the angular kinetic energy of the object, the formula is 1/2 (moment of inertia) (angular velocity)^2. The moment of inertia for a circle can be derived using the formula I = ∫r^2 dm or simply by taking the sum of r^2 multiplied by the mass of each point. This approach can also be used to calculate the angular kinetic energy.
  • #1
David112234
105
3
So the moment of inertia or a ring is MR2 I don't understand why. Here is my reasoning

Consider this shape (the ball is a point), the moment of inertia is MR2, there I agree

dumbell2.jpg


but now
what happens when you add another point on the other side

dumbell.jpg


since I = ΣMR2 then this is 2MR2

What about a ring or circle, that is nothing more than a bunch of points all the same distance from the center

dumbell3.jpg


so how many points are there? The circumference
so I= πR2MR2

I know that different objects moment of inertia differ by a number in front, like 1/2 or 3/5
Why is their no number prefix for Inertia in front of the formula for a hoop/ hollow cylinder/ circle that represents its circumference? How is this even derived?

Second question, How do you represent the angular Kinetic energy of this object?

dumbell.jpg
Left Point = A
Right point = B

½ I ωA2 + ⅓ I ωB2

½ I ( ωA2B2 )

both objects have same ω so

½ I 2ω2

Or

do I use ω to represent the angular velocity of the whole object and just keep it as

½ I ω2 with whatever I is from the previous question?
 
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  • #2
David112234 said:
since I = ΣMR2 then this is 2MR2
You have two point-like balls of mass M at distance R. That's a total mass of M=2M. I = MR2.
 
  • #3
but what about R, isn't their as many R's as there are points? Then where do the numbers in front of them moment of inertia come from? I how do you derive the MoI of a circle? Is my way correct? What about the second question?
 
  • #4
David112234 said:
but what about R, isn't their as many R's as there are points? Then where do the numbers in front of them moment of inertia come from? I how do you derive the MoI of a circle? Is my way correct? What about the second question?
For a collection mass or masses all at the same distance from the circle the "R" is the same for all.

For the second question... Just what is your second question?
 
  • #5
jbriggs444 said:
For a collection mass or masses all at the same distance from the circle the "R" is the same for all.

For the second question... Just what is your second question?
the second half of my post, the KE
 
  • #6
The second half of your post is confused. It mentions I without a formula for I. It uses ##\frac{1}{3} I \omega ^2## with no motivation. It mentions I using a single symbol for three distinct meanings.

I repeat: What is your question?
 
  • #7
jbriggs444 said:
The second half of your post is confused. It mentions I without a formula for I. It uses ##\frac{1}{3} I \omega ^2## with no motivation. It mentions I using a single symbol for three distinct meanings.

I repeat: What is your question?

thats supposed to be 1/2
The formula for angular KE is 1/2 (moment of Inertia) (angular velocity)^2

but I got confused about what the MOI for that shape is supposed to be, and how to represent the angular velocity

How would you write the KE of that shape?
here was my reasoning

½ I ωA2 + ⅓ I ωB2

½ I ( ωA2+ωB2 )

both objects have same ω so

½ I 2ω2

Or

do I use ω to represent the angular velocity of the whole object and just keep it as
 
  • #8
First label individually...(assuming a common axis of rotation)
Particle A has rotational-KE: ##K_{rot,A}=\frac{1}{2}I_A \omega_A^2=\frac{1}{2}(m_Ar_A^2) \omega_A^2##
Particle B has rotational-KE: ##K_{rot,B}=\frac{1}{2}I_B \omega_B^2=\frac{1}{2}(m_Br_B^2) \omega_B^2##
The rotational-KE of the system ##K_{rot}=K_{rot,A}+K_{rot,B}##
Since the system is rigidly rotating, ##\omega_A=\omega_B## (simply call them ##\omega##).
I'm sure you can complete the story.
 
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  • #9
David112234 said:
how do you derive the MoI of a circle?
[tex]I = \int r^2 dm[/tex]
source: https://en.wikipedia.org/wiki/Moment_of_inertia
David112234 said:
Is my way correct?
Of course. You are using the sum instead of the integral, which is correct for a simple case:
[tex]I = \sum r^2\Delta m = r_A^2 m_A + r_B^2 m_B[/tex]
if [itex]r_A = r_B = r[/itex] and [itex]m_A + m_B = m[/itex], then:
[tex]I = mr^2[/tex]
David112234 said:
What about the second question?
Same thing:
[tex]E = E_A + E_B = \frac{1}{2}I_A \omega_A^2 + \frac{1}{2}I_B \omega_B^2 = \frac{1}{2}r_A^2 m_A \omega_A^2 + \frac{1}{2}r_B^2 m_B \omega_B^2[/tex]
if [itex]r_A = r_B = r[/itex], [itex]m_A + m_B = m[/itex] and [itex]\omega_A= \omega_B= \omega[/itex], then:
[tex]E = \frac{1}{2}mr^2 \omega^2 = \frac{1}{2}I \omega^2[/tex]
where [itex]I = mr^2 = I_A + I_B[/itex]
 
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1. What is the difference between moment of inertia and angular kinetic energy?

The moment of inertia is a measure of an object's resistance to rotational motion, while angular kinetic energy is a measure of the energy that an object possesses due to its rotational motion.

2. How do you calculate the moment of inertia of a rigid body?

The moment of inertia of a rigid body can be calculated by multiplying the mass of each particle in the body by the square of its distance from the axis of rotation and then summing all of these values together.

3. Why is it important to understand moment of inertia and angular kinetic energy?

Understanding moment of inertia and angular kinetic energy is important in many fields of science and engineering, such as mechanics, robotics, and aerospace. These concepts are essential for predicting the behavior of rotating systems and designing efficient and stable structures.

4. Can the moment of inertia of an object change?

Yes, the moment of inertia of an object can change if its mass distribution or axis of rotation changes. For example, a figure skater can change their moment of inertia by pulling their arms in or extending them out while spinning.

5. What is the relationship between moment of inertia and angular velocity?

The moment of inertia and angular velocity of an object are directly proportional. This means that as the moment of inertia increases, the angular velocity decreases, and vice versa. This relationship is described by the equation Iω = L, where I is the moment of inertia, ω is the angular velocity, and L is the angular momentum.

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