Moment of Inertia: Solid vs Hollow Disk on Incline

In summary, the question asks which disk, a solid wood disk or a hollowed out disk of equal mass, would reach the bottom of an incline first when rolled down. The answer lies in the moment of inertia, where the hollow disk has a greater proportion of its mass located further away from the axis of rotation, resulting in a greater moment of inertia. This means that the solid disk would reach the bottom first due to its smaller moment of inertia.
  • #1
Invictus1017
2
0

Homework Statement


Alright, so say I have a solid wood disk, and a hollowed out disk of equal mass.
I roll them both down an incline, which one gets to the bottom first and why?
The scenario is very similar to this:
http://youtube.com/watch?v=7mxV6f5nuJY



Homework Equations


I = MR ?




The Attempt at a Solution


Something do with moment of inertia i think.

Thanks alot.
 
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  • #2
As you say, the moment of inertia is a crucial factor. You can answer this question quantitatively that is, explicitly calculate the moment of inertia for each disk and then evaluate it's acceleration. An alternative (and much easier) method would be to use the definition of Moment of Inertia for a point particle (I=mr2 not I=mr as you have above), and logical reasoning.

So to start we know that both their masses are equal, using the definition of I that I gave you above, can you make the next step?
 
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  • #3
I'm not sure but, the radius from the center of mass to the axis of the hollow disk is larger than the radius of the solid disk? Resulting in a smaller moment of inertia for the solid disk?
 
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  • #4
Invictus1017 said:
I'm not sure but, the radius from the center of mass to the axis of the hollow disk is larger than the radius of the solid disk? Resulting in a smaller moment of inertia for the solid disk?
Well you conclusion is correct, but your reasoning is wrong. The centre of mass of both disc both lie on the axis of rotation. However, the hollow disc has a greater proportion of its mass located further away from the axis of rotation, thus the moment of inertia is greater. Does that make sense?
 

Related to Moment of Inertia: Solid vs Hollow Disk on Incline

What is moment of inertia and how does it differ between solid and hollow disks?

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution of the object and its distance from the axis of rotation. For a solid disk, the mass is evenly distributed and the moment of inertia is given by 1/2 * MR^2, where M is the mass of the disk and R is the radius. For a hollow disk, the mass is concentrated at the edges and the moment of inertia is given by 1/2 * MR^2, where M is the mass of the disk and R is the radius of the outer edge.

What is the difference in the moment of inertia between a solid and hollow disk on an incline?

The moment of inertia of a solid disk on an incline will be greater than that of a hollow disk due to the distribution of mass. The solid disk has mass distributed throughout its entire radius, while the hollow disk has most of its mass concentrated at the edges. This means that the solid disk will have a greater resistance to changes in its rotational motion compared to the hollow disk on an incline.

How does the moment of inertia affect the rotational motion of a solid and hollow disk on an incline?

The moment of inertia plays a crucial role in determining the rotational motion of an object. A higher moment of inertia means the object will require more torque to rotate, and will have a slower angular acceleration. This means that the solid disk will have a slower rotational motion compared to the hollow disk on an incline, due to its higher moment of inertia.

Is there a real-life application of understanding the moment of inertia of solid and hollow disks on an incline?

Yes, the concept of moment of inertia is important in many real-life applications, such as designing machines with rotating parts, analyzing the stability of structures, and understanding the dynamics of sports equipment. In the case of solid and hollow disks on an incline, understanding the moment of inertia can help engineers in designing more efficient and stable machines and structures.

How can the moment of inertia be calculated for a solid and hollow disk on an incline?

The moment of inertia for a solid disk on an incline can be calculated using the formula 1/2 * MR^2, where M is the mass of the disk and R is the radius. For a hollow disk on an incline, the moment of inertia can be calculated using the formula 1/2 * MR^2, where M is the mass of the disk and R is the radius of the outer edge. These calculations can be done using a variety of tools, such as mathematical equations, computer simulations, or physical experiments.

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