Moment of inertia of a rotated line

In summary, the problem involves finding the moment of inertia (MOI) for a drum that consists of a cylinder and a rotated line. The MOI for the cylinder is 1/2*M*(a^2+b^2) where a and b are the inner and outer radii, respectively. For the line, the MOI is given by 1/2*rho*pi*int(0, h){(mx+c)^4 - c^4} dx, where h is the height of the drum and m and c are the slope and y-intercept of the line. The integration limits should be from 0 to the height of the drum, not from x1 to x2. It is important to note
  • #1
rock.freak667
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Homework Statement


I have a drum that comprises of basically a cylinder and rotated line in the form y=mx+c

http://img22.imageshack.us/img22/6108/druma.jpg [Broken]


Homework Equations



[tex]I=\int r^2 dm[/tex]

The Attempt at a Solution



I found the MOI for the cylinder as 1/2M(a2+b2) where a and b are the inner and outer radii respectively.

I am getting trouble forming the MOI for the line. I am supposed to consider an elemental section, but I am not sure if I should consider a disc or some other shape.

Disc: (Width dx and radius 'y')

dI=1/2(dm)y2

dm=ρ dV = ρ(πy2dx)

dI=1/2ρπ y4 dx

[tex]I_{line}= \frac{1}{2} \rho \pi \int_{x_1} ^{x_2} y^4 dx[/tex]


Since I have two lines, the inertia would be


[tex]I = \frac{1}{2} \rho \pi \int_{x_1} ^{x_2} (y_2^4 -y_1^4) dx[/tex]

I can appropriately select x2 and x1 based on height of the cylinder and the height of the total drum. I can find the ρ for the material. But I'd need to calculate the equations for y2 and y1.

So my MOI would be

[tex]I = \frac{1}{2} M(a^2+b^2)+ \frac{1}{2} \rho \pi \int_{x_1} ^{x_2} (y_2^4 -y_1^4) dx[/tex]

Is this correct or did I make a mistake in my element or in the algebra?
 
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  • #2


Your approach seems to be on the right track. The equation for the MOI of the line is correct, but the integration limits should be from 0 to the height of the drum, not from x1 to x2. Also, the equation for y should be in terms of x, not y. So the final equation for the MOI would be:

I = 1/2*M*(a^2+b^2) + 1/2*rho*pi*int(0, h){(mx+c)^4 - c^4} dx

Where h is the height of the drum and m and c are the slope and y-intercept of the line, respectively.

I hope this helps. Good luck with your calculations!
 

What is the moment of inertia of a rotated line?

The moment of inertia of a rotated line is a measure of an object's resistance to rotational motion around a specific axis. It is the rotational equivalent of mass in linear motion and is affected by the distribution of mass around the axis of rotation.

How is the moment of inertia of a rotated line calculated?

The moment of inertia of a rotated line can be calculated using the formula I = MR^2, where I is the moment of inertia, M is the mass of the object, and R is the distance of the object from the axis of rotation. This formula assumes that the line is a thin, straight rod with all the mass concentrated at the center.

What factors affect the moment of inertia of a rotated line?

The moment of inertia of a rotated line is affected by the mass and distribution of mass around the axis of rotation. It also depends on the shape and size of the object, as well as the axis of rotation. A longer line will have a greater moment of inertia than a shorter line with the same mass.

Why is the moment of inertia important?

The moment of inertia is important because it determines how easily an object can be rotated around a specific axis. It is crucial in understanding and predicting an object's rotational motion, such as how fast it will spin or how much energy is required to rotate it.

How is the moment of inertia of a rotated line used in real-life applications?

The moment of inertia of a rotated line is used in various real-life applications, such as designing machines and structures that involve rotational motion, such as motors, propellers, and bridges. It is also used in sports, such as figure skating and gymnastics, to understand and improve the performance of athletes' movements.

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