Moment of intertia for rod with weight on one end

In summary, the conversation is about deriving the moment of inertia for a long rod with a point-weight on one end. The formula for the rod is normally (1/2)ML^2, and it is suggested to add the moment of inertia of the point mass to that of the rod to get the moment of inertia of the entire object. It is clarified that the moment of inertia of a rod about its center is (1/12)ML^2. The conversation then discusses the specifics of the rod's length and mass, and it is determined that the moment of inertia of the weight on the end is 31.09295 and the total moment of inertia is 52.1017.
  • #1
smegmaster
4
0
Hello,

I'm trying to derive the moment of inertia for a long rod with a point-weight on one end (the rod rotates through its midpoint). Could anyone offer any help? I know the formula for the rod is normally (1/2)ML^2, but I'm not sure how the weight on one end alters that.

Thanks!
 
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  • #2
Add the moment of inertia of the point mass to that of the rod; that will give you the moment of inertia of the entire object.

FYI: I assume you meant that the moment of inertia of a rod about its center is (1/12)ML^2.
 
  • #3
Doc Al said:
Add the moment of inertia of the point mass to that of the rod; that will give you the moment of inertia of the entire object.

FYI: I assume you meant that the moment of inertia of a rod about its center is (1/12)ML^2.

1/12, right. So I can just add the mass of the extra weight to the rod's moment of intertia?

Thanks for the help.
 
  • #4
The length of the rod I have is 4.9m, with its mass 10.5kg. Without the weight, the moment of inertial is 21.00875. If the mass of the weight on the end is 5.18kg, would I be correct in saying the new moment of inertia is just 21.00875 + 5.18 = 26.18875kg?
 
  • #5
smegmaster said:
So I can just add the mass of the extra weight to the rod's moment of intertia?
No. (For one thing, mass and moment of inertia are different kinds of quantities--with different units--so you can't add them.) Add the moment of inertia of the point mass to the rod's moment of inertia. (What's the moment of inertia of a mass about some point?)
 
  • #6
Ahh, sorry. Moment of inertia for a point is its mass times the square of the radius. Thus the moment of inertia for the weight is 5.18*(4.9/2)^2 = 31.09295, bringing the total moment of inertia to 52.1017. Thanks!
 

What is the moment of inertia for a rod with weight on one end?

The moment of inertia for a rod with weight on one end is a measure of the object's resistance to changes in rotational motion. It is represented by the symbol "I" and has units of kg*m^2. It is calculated by taking the mass of the object and multiplying it by the square of the distance from the axis of rotation to the object's center of mass.

How does the position of the weight affect the moment of inertia for a rod?

The position of the weight on the rod does affect the moment of inertia. The further the weight is from the axis of rotation, the larger the moment of inertia will be. This is because the weight has a greater lever arm and thus contributes more to the object's resistance to rotational motion.

What is the equation for calculating the moment of inertia for a rod with weight on one end?

The equation for calculating the moment of inertia for a rod with weight on one end is I = mL^2, where m is the mass of the object and L is the distance from the axis of rotation to the object's center of mass. This equation assumes that the object is a thin, uniform rod with the weight located at the end.

How does the moment of inertia for a rod with weight on one end compare to a rod with weight distributed along its length?

The moment of inertia for a rod with weight on one end will be greater than a rod with weight distributed along its length, assuming all other factors are equal. This is because the weight on one end has a longer lever arm and thus contributes more to the object's resistance to rotational motion.

What are some real-world applications of understanding moment of inertia for a rod with weight on one end?

Understanding the moment of inertia for a rod with weight on one end is important in various fields such as engineering, physics, and sports. It helps engineers design more stable structures, physicists predict the motion of objects, and athletes improve their performance in sports like gymnastics and figure skating.

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