How does the moment of inertia scale?

In summary, the conversation discusses the effects of multiplying design dimensions by a scaling factor on the volume, mass, and moment of inertia of an object. The moment of inertia is given by the equation I=k*m*r^2, where k is a constant that depends on the shape. By multiplying each quantity by the scale factor, f, the mass and volume will be multiplied by f^3, and the moment of inertia will be multiplied by a constant factor, depending on the shape.
  • #1
student34
639
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Homework Statement



If we multiply all of the design dimensions by a scaling factor f, it's volume and mass will be multiplied by f^3. (a) By what factor will its moment of inertia be multiplied? (b) And if a 1/48 scale model has a rotational kinetic energy of 2.5J, what will be the kinetic energy for the full-scale object of the same material rotating at the same angular velocity?

Homework Equations



The Attempt at a Solution



I have absolutely no idea what they are talking about. I feel like I missed an entire chapter, but I am sure that I didn't. All the other questions in my book leading up to this one have been the usual ones that ask me to find a value when dealing with the moment of inertia. What is this asking.
 
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  • #2
Scale Factors

If you look at some of the moment of inertia equations you know, you'll see how we can address this question.

For a point mass, it's [itex]I = \frac{2 m r^2}{5} [/itex]. For a solid sphere, [itex]I = \frac{2 m r^2}{5} [/itex]. For a hollow sphere, [itex] I = \frac{2 m r^2}{3} [/itex]. Rod about the end [itex]I = \frac{1 m r^2}{3} [/itex]. Rod about the middle: [itex]I = \frac{1 m r^2}{12} [/itex].

The only thing that is changing is the constant in front.

The moment of inertia is given by the equation [itex]I=k*m*r^2[/itex], where k is the constant that depends on the shape. So the moment increasesas the mass increases, and increases like the square of the radius.

To find the scaling, you just multiply each quantity by a scale factor, f. We have to multiply by f3 for mass (since mass increases with increasing *volume*). We multiply each term of r by f.

Hope that helps.

Dr Peter Vaughan
BASIS Peoria Physics
 
  • #3
sonnyfab said:
If you look at some of the moment of inertia equations you know, you'll see how we can address this question.

For a point mass, it's [itex]I = \frac{2 m r^2}{5} [/itex]. For a solid sphere, [itex]I = \frac{2 m r^2}{5} [/itex]. For a hollow sphere, [itex] I = \frac{2 m r^2}{3} [/itex]. Rod about the end [itex]I = \frac{1 m r^2}{3} [/itex]. Rod about the middle: [itex]I = \frac{1 m r^2}{12} [/itex].

The only thing that is changing is the constant in front.

The moment of inertia is given by the equation [itex]I=k*m*r^2[/itex], where k is the constant that depends on the shape. So the moment increasesas the mass increases, and increases like the square of the radius.

To find the scaling, you just multiply each quantity by a scale factor, f. We have to multiply by f3 for mass (since mass increases with increasing *volume*). We multiply each term of r by f.

Hope that helps.

Dr Peter Vaughan
BASIS Peoria Physics

Oh, thank-you so much!
 

1. How does the moment of inertia scale with mass?

The moment of inertia is directly proportional to the mass of an object. This means that as the mass of an object increases, its moment of inertia will also increase. This relationship can be described by the equation I = mr², where I is the moment of inertia, m is the mass, and r is the distance from the axis of rotation.

2. How does the moment of inertia scale with distance from the axis of rotation?

The moment of inertia is directly proportional to the square of the distance from the axis of rotation. This means that as the distance from the axis of rotation increases, the moment of inertia will increase at a faster rate. This relationship can be described by the equation I = mr², where I is the moment of inertia, m is the mass, and r is the distance from the axis of rotation.

3. How does the moment of inertia scale with shape?

The moment of inertia is affected by the shape of an object. Objects with a larger radius of gyration, or a greater distribution of mass from the axis of rotation, will have a larger moment of inertia. This means that objects with a longer or more spread out shape will have a larger moment of inertia than objects with a compact or concentrated shape.

4. How does the moment of inertia scale with rotation axis?

The moment of inertia is dependent on the axis of rotation. The moment of inertia will be different for the same object depending on the axis it is rotated around. For example, a ring will have a different moment of inertia when rotated around its diameter compared to when it is rotated around its center axis.

5. How does the moment of inertia scale with angular velocity?

The moment of inertia does not directly scale with angular velocity. However, the moment of inertia does affect an object's angular velocity. When a force is applied to an object, the moment of inertia will determine how quickly the object will rotate. Objects with a larger moment of inertia will have a slower angular velocity than objects with a smaller moment of inertia.

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