Apologies for starting yet another moment of inertia thread.
I had hoped that the moi overview thread would be an educational thread with respect to helping people understand moi generically and that the moi and the cast thread could be about discussing how and if moi applies to the cast.
This being Sexyloops and the Internet people wandered from the moi and the cast thread to the moi overview thread. Fwiw this isn’t a condemnation of that activity. The discussions are interesting and, I think, worthwhile but, I did have some requests to provide just an educational overview of moi so I will try again.
I’m hoping that this thread can be educational in nature only, i.e. what is moment of inertia, how do we calculate it and why do we care and that the other threads can be used for discussing moi and the cast specifically.
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MOI again
Moderator: Torsten
Re: MOI again
I’ll start with an overview of Newton’s three laws, often referred to as N1, N2 and N3, and then discuss their rotational equivalents.
Newton’s First Law of Motion states that an object at rest tends stay at rest and an object in a state of uniform motion, i.e. moving at constant
speed and direction, tends to stay in uniform motion unless an external force acts on it. This is also known as the Law of Inertia. Inertia is the name for the tendency of an object to remain at rest or in uniform motion without an external force acting on it. Inertia is a measure of an object’s resistance to change in motion in response to an applied force. Note that the object does not have to be in motion to have inertia. Also, an object’s inertia does not change whether it is in motion or at rest. We measure the amount of inertia that an object has by its mass. A 2 kg mass has twice the resistance to change due an applied force that a 1 kg mass has.
Newton’s Second Law can be stated as Force = mass x acceleration (F=ma). It predicts how an object’s motion will change given its mass and an applied force. Note that force and acceleration are vector quantities. They have both magnitude and direction. This means that when I throw a ball due north it will accelerate and travel due north unless there are other forces acting on it. It also means that if I have multiple forces acting on an object that I can add the components of the individual forces together to determine the net force to use in the F=ma equation.
Newton’s Third Law states that for every action there is an equal and opposite reaction. This means that when you push on an object with a given amount of force it pushes back with exactly the same amount of force but in the opposite direction. It’s easy to think that when I push a marble that I must be pushing it harder than it is pushing back because the marble moves but I don’t. That isn’t correct. The marble is pushing back with exactly the same force but it has much less mass so it will accelerate much more than you will. N3 is the reason that when you throw a 1 kg ball with the same acceleration as a 2 kg ball the 2 kg ball feels harder to throw or heavier.
Newton’s laws apply to any object but for simplicity we usually talk about point masses and non-flexible objects.
Another simplification we often use when talking about Newton’s laws is that we ignore gravity and friction. These forces are always present in the real world and for a complete analysis of an object’s motion we usually must include these. When I throw a ball, for example, ignoring external forces like gravity and air resistance the ball will travel forever. If we want to determine the optimal launch angle for maximum distance when we include gravity but ignore fiction we find that the optimal angle to throw the ball is 45 degrees above horizontal. If we include both gravity and air resistance we find that the optimal angle will be a bit less than 45 degrees. Finding the optimal angle will depend on the initial speed of the ball when it leaves our hand because the air resistance to the balls movement varies with square of the velocity of the ball. As the ball slows down the air resistance decreases.
Newton’s First Law of Motion states that an object at rest tends stay at rest and an object in a state of uniform motion, i.e. moving at constant
speed and direction, tends to stay in uniform motion unless an external force acts on it. This is also known as the Law of Inertia. Inertia is the name for the tendency of an object to remain at rest or in uniform motion without an external force acting on it. Inertia is a measure of an object’s resistance to change in motion in response to an applied force. Note that the object does not have to be in motion to have inertia. Also, an object’s inertia does not change whether it is in motion or at rest. We measure the amount of inertia that an object has by its mass. A 2 kg mass has twice the resistance to change due an applied force that a 1 kg mass has.
Newton’s Second Law can be stated as Force = mass x acceleration (F=ma). It predicts how an object’s motion will change given its mass and an applied force. Note that force and acceleration are vector quantities. They have both magnitude and direction. This means that when I throw a ball due north it will accelerate and travel due north unless there are other forces acting on it. It also means that if I have multiple forces acting on an object that I can add the components of the individual forces together to determine the net force to use in the F=ma equation.
Newton’s Third Law states that for every action there is an equal and opposite reaction. This means that when you push on an object with a given amount of force it pushes back with exactly the same amount of force but in the opposite direction. It’s easy to think that when I push a marble that I must be pushing it harder than it is pushing back because the marble moves but I don’t. That isn’t correct. The marble is pushing back with exactly the same force but it has much less mass so it will accelerate much more than you will. N3 is the reason that when you throw a 1 kg ball with the same acceleration as a 2 kg ball the 2 kg ball feels harder to throw or heavier.
Newton’s laws apply to any object but for simplicity we usually talk about point masses and non-flexible objects.
Another simplification we often use when talking about Newton’s laws is that we ignore gravity and friction. These forces are always present in the real world and for a complete analysis of an object’s motion we usually must include these. When I throw a ball, for example, ignoring external forces like gravity and air resistance the ball will travel forever. If we want to determine the optimal launch angle for maximum distance when we include gravity but ignore fiction we find that the optimal angle to throw the ball is 45 degrees above horizontal. If we include both gravity and air resistance we find that the optimal angle will be a bit less than 45 degrees. Finding the optimal angle will depend on the initial speed of the ball when it leaves our hand because the air resistance to the balls movement varies with square of the velocity of the ball. As the ball slows down the air resistance decreases.
"There can be only one." - The Highlander. 
Physics for physics sake. Faith for casting sake.

Physics for physics sake. Faith for casting sake.
Re: MOI again
In linear systems inertia is the tendency of an object to remain in motion or stay at rest unless a force causes their speed or direction to change. An object’s inertia is directly proportional to its mass. An object has inertia whether it is moving or stationary. An object’s inertia does not change if it is moving or stationary. It only changes if I add or remove mass from the object.
Torque is a measure of the force that can cause an object to rotate about an axis. If I apply a torque to an object, I am compelling it to rotate about an axis.
Moment of inertia is the tendency of an object to remain in angular motion or at rest unless a torque causes their angular speed to change or direction. An object’s moment of inertia is directly related to its mass and the square of the distance from the object to some arbitrarily selected point. The object’s moi does not change whether it is moving or stationary.
Consider the following point mass system with Mass A and Mass B. Both have the same inertia because they have the same mass. They have the same axis of rotation but their moi values are significantly different because of their distance from that axis of rotation. Since they have the same axis of rotation we can determine their combined moi by simply adding them together.
We can use this additive property to determine the moi of differently shaped objects. By breaking up the object into minute parts, calculating their individual moi relative to our chosen axis of rotation and then summing the individual moi values we can determine the moi of the complete object.
In the case of a uniform rod of length L and mass M where the mass is uniformly distributed along the rod we can determine that the moi of the rod relative to an axis of rotation at its midpoint is (ML^2)/12. If we change the axis of rotation to one of the endpoints of the rod then the formula for its moi becomes (ML^2)/3
Torque is a measure of the force that can cause an object to rotate about an axis. If I apply a torque to an object, I am compelling it to rotate about an axis.
Moment of inertia is the tendency of an object to remain in angular motion or at rest unless a torque causes their angular speed to change or direction. An object’s moment of inertia is directly related to its mass and the square of the distance from the object to some arbitrarily selected point. The object’s moi does not change whether it is moving or stationary.
Consider the following point mass system with Mass A and Mass B. Both have the same inertia because they have the same mass. They have the same axis of rotation but their moi values are significantly different because of their distance from that axis of rotation. Since they have the same axis of rotation we can determine their combined moi by simply adding them together.
We can use this additive property to determine the moi of differently shaped objects. By breaking up the object into minute parts, calculating their individual moi relative to our chosen axis of rotation and then summing the individual moi values we can determine the moi of the complete object.
In the case of a uniform rod of length L and mass M where the mass is uniformly distributed along the rod we can determine that the moi of the rod relative to an axis of rotation at its midpoint is (ML^2)/12. If we change the axis of rotation to one of the endpoints of the rod then the formula for its moi becomes (ML^2)/3
"There can be only one." - The Highlander. 
Physics for physics sake. Faith for casting sake.

Physics for physics sake. Faith for casting sake.
Re: MOI again
If we can determine the moi of an object about an axis through it’s center of mass (centroid) we can use the Parallel Axis Theorem to determine its center of mass through any axis of rotation that is parallel to that axis.
Moi about the parallel axis = moi about center of mass + M d^2
Where:
M is the mass of the object
d is the distance from the center of mass to the new center of mass
Using our previous example of the rod we determined that its moi about it’s center of mass was ML2/12. If we want to know the moi around one end of the rod then the distance from the center of mass d is L/2 so our moi around the end point can determined with the parallel axis theorem by
Moi about center + M d^2
= (ML^2)/12 + (ML^2)/4
= (ML^2)/3
Moi about the parallel axis = moi about center of mass + M d^2
Where:
M is the mass of the object
d is the distance from the center of mass to the new center of mass
Using our previous example of the rod we determined that its moi about it’s center of mass was ML2/12. If we want to know the moi around one end of the rod then the distance from the center of mass d is L/2 so our moi around the end point can determined with the parallel axis theorem by
Moi about center + M d^2
= (ML^2)/12 + (ML^2)/4
= (ML^2)/3
"There can be only one." - The Highlander. 
Physics for physics sake. Faith for casting sake.

Physics for physics sake. Faith for casting sake.
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- Posts: 1189
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Re: MOI again
Walter,
Thanks for the synopsis. My grey cells are greying faster than my hair. You covered about half of one of my semesters of intro physics in 4 posts!
Thinking about moi took me back to those days. I had made a friend who not only shared some of my science classes, but also had a similar passion for frisbees. We both also had another somewhat niche passion: their's was archery, mine was fly casting.
I showed them mine and they did the same.
I remember that their bow was much more sophisticated than usual, but since this was decades ago, there were no pulleys involved, and there was only one stabilizer, if I remember correctly. Today, archery equipment looks like it comes from outer space!
Anyway... I remember being surprised that when the arrow had left their bow, the bow would rotate forward around their relaxed grip, every time. I do not remember if either of us said it out loud, or if I just imagined it, but I remember thinking that the system must have induced a moi in the top section of the bow around the axis of their grip?
I'm not sure how to explain that. Maybe I'm wrong?
But, I kinda saw the top of the bow being a lot like a fly rod. Even today, when I do not intentionally try to apply a lot of torque, I feel the rod continue to rotate after loop launch, I suspect something similar is taking place???
Gary
Thanks for the synopsis. My grey cells are greying faster than my hair. You covered about half of one of my semesters of intro physics in 4 posts!
Thinking about moi took me back to those days. I had made a friend who not only shared some of my science classes, but also had a similar passion for frisbees. We both also had another somewhat niche passion: their's was archery, mine was fly casting.
I showed them mine and they did the same.
I remember that their bow was much more sophisticated than usual, but since this was decades ago, there were no pulleys involved, and there was only one stabilizer, if I remember correctly. Today, archery equipment looks like it comes from outer space!
Anyway... I remember being surprised that when the arrow had left their bow, the bow would rotate forward around their relaxed grip, every time. I do not remember if either of us said it out loud, or if I just imagined it, but I remember thinking that the system must have induced a moi in the top section of the bow around the axis of their grip?
I'm not sure how to explain that. Maybe I'm wrong?
But, I kinda saw the top of the bow being a lot like a fly rod. Even today, when I do not intentionally try to apply a lot of torque, I feel the rod continue to rotate after loop launch, I suspect something similar is taking place???
Gary
With appreciation and apologies to Ray Charles…
“If it wasn’t for AI, we wouldn’t have no I at all.”
“If it wasn’t for AI, we wouldn’t have no I at all.”
Re: MOI again
Gary,
Was it like this or more pronounced?
Was it like this or more pronounced?
"There can be only one." - The Highlander. 
Physics for physics sake. Faith for casting sake.

Physics for physics sake. Faith for casting sake.
-
- Posts: 1189
- Joined: Tue Jan 29, 2013 7:51 am
Re: MOI again
Walter,
Yeah, that looks to be about right. But this was many years ago, the bow was wood, and there was only one short extension out front of the bow. And I was watching in real time... so, possibly, it might have been a bit more pronounced.
I also got the impression that my friend actually enjoyed the experience, and allowed it to happen to its fullest.
Gary
Yeah, that looks to be about right. But this was many years ago, the bow was wood, and there was only one short extension out front of the bow. And I was watching in real time... so, possibly, it might have been a bit more pronounced.
I also got the impression that my friend actually enjoyed the experience, and allowed it to happen to its fullest.
Gary
With appreciation and apologies to Ray Charles…
“If it wasn’t for AI, we wouldn’t have no I at all.”
“If it wasn’t for AI, we wouldn’t have no I at all.”