Moment of Inertia Help for a Bike Wheel Problem

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In summary, the moments of inertia of a wheel formed from a hoop and equally spaced spokes can be calculated by breaking it down into two steps: one for the hoop and one for the rods. The moment of inertia for the hoop is equal to the mass of the hoop times the square of the radius. For the rods, the parallel-axis theorem must be used and the moment of inertia is equal to the moment of inertia of a long, thin rod plus the distance from the center of mass squared times the mass of the rod. The total moment of inertia for the wheel is then calculated by adding the two systems together. The axis of rotation is perpendicular to the entire wheel and the coordinates can be set up in any way, with the z
  • #1
sirfederation
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Moments of inertia Help

A wheel is formed from a hoop and n equally spaced spokes extending from the center of the hoop to its rim. The mass of the hoop is M, and the radius of the hoop (and hence the length of each spoke) is R. The mass of each spoke is lower case m.

A. Determine the moments of inertia of the wheel about an axis through its center and perpendicular to the plane of the wheel.

From a table in my book the long thin rod with rotation axis through the center is Icm = (1/12)m(L)^2

I think I have to do this problem in two steps, one for the hoop and one for the rods.
(I think the moment of inertia is about the z axis through the origin)

Iz = integral (r^2) dm = (R^2) integral dm =M(R^2)

Ok, for the rods I have to apply the parallel-axis theorem
I = Icm+M(D^2)
I = (1/12)m(R^2) + m(R^2)
I = n ((1/12)M(R^2) + m (R^2))

So now I just add the two systems together to find the moment of inertia.

I = n ((1/12)m(R^2) + m (R^2))+ M(R^2)

Ok, here is what I am confused about: I am sure that the hoop passes through the z axis but I do not think the rods do. I think the rods pass through the y-axis because it says perpendicular to the wheel which is the hoop plus the rods. If that is true then, my answer changes to I = n(1/12)m(R^2)+ M(R^2). Can you actually have two objects passing thorough two different axis?

B. Determine the moment of inertia of the wheel about an exis through its rim and perpendicular to the plane of the wheel.

I still need to finish this one please do not reply to B until I put what I got. thanks
 
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  • #2
Ok, here is what I am confused about : I am sure that the hoop passes through the z axis but I do not think the rods do...Can you actually have two objects passing thorough two different axis?
What do you mean by "passing through" an axis? The axis of rotation is perpendicular to the entire wheel. The rim of the wheel does not pass through the axis of rotation but the spokes do- exactly the opposite of what you say.
As far as the x,y, z axes are concerned- you are free to set up the coordinate axes as you wish. I assume that you set up your coordinate system so that the wheel is in the xy-plane and the z-axis goes through the center of the wheel. In that case, yes, it is distance from the origin that is important.
 
  • #3
Here is my final answer for A.
A. I = M(R^2)+N(( m(R^2) )/3)

Here is my final answer for B.

Irim= I + (M + nm)(R^2)
Irim= M(R^2)+N(( m(R^2) )/3)+ (M + nm)(R^2)
 
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1. What is the moment of inertia of a bike wheel?

The moment of inertia of a bike wheel is a measure of its resistance to changes in its rotational motion. It takes into account the mass, shape, and distribution of mass of the wheel.

2. How is the moment of inertia calculated for a bike wheel?

The moment of inertia for a bike wheel can be calculated using the formula I = mr^2, where I is the moment of inertia, m is the mass of the wheel, and r is the radius of the wheel.

3. Why is the moment of inertia important for a bike wheel?

The moment of inertia is important for a bike wheel because it affects the wheel's rotational stability and how it responds to external forces, such as when turning or going over bumps. A higher moment of inertia means the wheel will be harder to accelerate or change directions, while a lower moment of inertia allows for quicker changes in speed or direction.

4. How can the moment of inertia be changed for a bike wheel?

The moment of inertia of a bike wheel can be changed by altering the distribution of mass. This can be done by adding or removing weight from different parts of the wheel or by changing the shape of the wheel. For example, a wheel with a larger diameter will have a higher moment of inertia than a wheel with a smaller diameter.

5. How can understanding the moment of inertia help with a bike wheel problem?

Understanding the moment of inertia can help with a bike wheel problem by allowing you to make informed decisions about how to improve the wheel's performance. For example, if the wheel is too difficult to turn, you may want to decrease the moment of inertia by removing weight from the rim. On the other hand, if the wheel is too responsive and unstable, you may want to increase the moment of inertia by adding weight to the rim.

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