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Problems calculating the New Moment of Inertia
- Thread starter KMconcordia
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In summary, the conversation is discussing different approaches to solving a problem involving conservation of angular momentum and linear momentum. The problem involves a disk and a mass attached to it, and the question is how to calculate the moment of inertia when the attached mass is added. One option is to find the mass centre of the system and use conservation of angular momentum, while another is to use conservation of angular momentum around a fixed point and also consider conservation of linear momentum. The conversation also discusses the importance of considering the mass of the attached object in the calculation. A method is suggested involving finding the new centre of mass of the system and then using conservation of angular momentum.
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There are a couple of ways to approach this problem.
You can find the mass centre of the system then use that angular momentum will be conserved about that point. I tend to avoid finding mass centres of systems because there is usually an easier way, but in this case it might be simplest.
Alternatively, you can take conservation of angular momentum about any fixed point - the initial position of either the centre of the disc or the point where the mass is attached, say. This way, you can just treat each body as contributing to angular momentum, but you will also need to use conservation of linear momentum. E.g. if you take the initial position of the disc's centre as axis then the subsequent linear motion of the attached particle needs to be related to the new angular rotation rate.
Your calculation of the new MoI is in respect of an axis at the perimeter of the disc. That would be fine if that point were the new centre of rotation, but it will not be. The mass of the attached object must feature in the answer. E.g. if the attached mass is very small there will be little change to the rotation of the disk.
You can find the mass centre of the system then use that angular momentum will be conserved about that point. I tend to avoid finding mass centres of systems because there is usually an easier way, but in this case it might be simplest.
Alternatively, you can take conservation of angular momentum about any fixed point - the initial position of either the centre of the disc or the point where the mass is attached, say. This way, you can just treat each body as contributing to angular momentum, but you will also need to use conservation of linear momentum. E.g. if you take the initial position of the disc's centre as axis then the subsequent linear motion of the attached particle needs to be related to the new angular rotation rate.
Your calculation of the new MoI is in respect of an axis at the perimeter of the disc. That would be fine if that point were the new centre of rotation, but it will not be. The mass of the attached object must feature in the answer. E.g. if the attached mass is very small there will be little change to the rotation of the disk.
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KMconcordia
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So which formula would be the correct way to approach part B?
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It's not a matter of some handy formula readily applied. You must first decide your approach. I described two options.KMconcordia said:
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Delta2
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@haruspex you say something about using conservation of linear momentum... I tried to read the problem as carefully as I could but I couldn't find any data regarding the linear momentum of any of the bodies, isn't the linear momentum 0 for all bodies of the system (before and after the "collision")?
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The system has zero linear momentum throughout, but after the collision they will each have linear momentum.Delta2 said:
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KMconcordia
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Do you have any other way in mind about approaching this problem? My final exams tomorrow and I am very lost..Delta2 said:
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Delta2
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Calculate the C.O.M of the disk+particle, then find the moment of inertia ##I'## of disk+particle around the C.O.M, then apply conservation of angular momentum ##I'\omega'=I\omega##
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Delta2
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b) would be approximately correct if the mass of 10kg was small compared to the mass of the disc so that the new center of rotation was about the same as the old center of rotation and the same as the center of the disc. But 10kg is a lot compared to the 0.5kg of the mass of the disc so this method isn't correct all.
As i said before you have to find the new c.o.m of the system which will be the new center of rotation. That's the tricky part of this exercise, that the center of rotation changes (and the center of mass as well) when the mass of 10kg is added to the disc.
To find the new c.o.m of the system: Imagine that you have a mass of 0.5kg as a point particle at the center, and the mass of 10kg at distance R=0.25m, then the c.o.m is at distance x from the center such that ##0.5x-10(R-x)=0##. Then calculate the total moment of inertia around the new c.o.m and then use conservation of angular momentum.
EDIT: I had a small mistake in the equation regarding the new c.o.m of the system which i corrected.
As i said before you have to find the new c.o.m of the system which will be the new center of rotation. That's the tricky part of this exercise, that the center of rotation changes (and the center of mass as well) when the mass of 10kg is added to the disc.
To find the new c.o.m of the system: Imagine that you have a mass of 0.5kg as a point particle at the center, and the mass of 10kg at distance R=0.25m, then the c.o.m is at distance x from the center such that ##0.5x-10(R-x)=0##. Then calculate the total moment of inertia around the new c.o.m and then use conservation of angular momentum.
EDIT: I had a small mistake in the equation regarding the new c.o.m of the system which i corrected.
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FAQ: Problems calculating the New Moment of Inertia
What is moment of inertia?
Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass and distribution of mass of the object.
Why is calculating the new moment of inertia important?
Calculating the new moment of inertia is important because it allows us to understand and predict how an object will behave when it is rotating. This information is crucial in designing and analyzing various mechanical systems.
What information is needed to calculate the new moment of inertia?
To calculate the new moment of inertia, we need to know the mass and distribution of mass of the object, as well as its shape and orientation in space.
What are some common problems encountered when calculating the new moment of inertia?
Some common problems include incorrect measurements of mass, incorrect assumptions about the distribution of mass, and not considering the orientation of the object in space.
How can these problems be avoided?
To avoid these problems, it is important to carefully measure the mass of the object and to consider all dimensions and orientations when determining its distribution of mass. It is also helpful to double check all calculations and assumptions to ensure accuracy.