Moment of Inertia of Wire Shaped like Astroid: Finding MOE | Solution Attempt

In summary, the problem is to find the moment of inertia of a wire shaped like the astroid x=cos3(t), y=sin3(t), t[0, 2*pi] with constant density k. The formula for the MOI is derived by integrating k* (x^2 + y^2)ds along the curve, where ds = 3*|cos(t)*sin(t)| dt. However, the mistake in the solution lies in forgetting the absolute value sign, resulting in the two contributions cancelling each other out and giving a final expression of zero. To correct this, it is suggested to integrate only one quarter of the astroid and quadruple the result, or express the MOI as C*M*(
  • #1
kasse
384
1

Homework Statement



A wire is shaped like the astroid x=cos3(t), y=sin3(t), t[0, 2*pi] and has constant density = k. Find its moment of intertia I0 around the origin.

2. The attempt at a solution

To find the MOE we must integrate k* (x2 + y2)ds along the curve. We differentiate and find that ds can be written 3*cos(t)*sin(t).The final expression is:

3k * Int (cos7(t)*sin(t) + sin7(t)*cos(t))dt for t [0, 2*pi]. But this is zero! What/where is my mistake?
 
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  • #2
Be careful when integrating periodically symmetric functions through a full period: your dt cycles through both positive and negative signs. You may be better off integrating only one quarter of the astroid and quadrupling that result. (Were you to express the moment of inertia as C · M · (R^2) , the moment for each quadrant would have the same constant C as the entire figure.)
 
  • #3
kasse said:


3k * Int (cos7(t)*sin(t) + sin7(t)*cos(t))dt for t [0, 2*pi]. But this is zero! What/where is my mistake?


Your mistake is: changing the function for ds when doing your simplification!

[tex]
ds = \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2} \;dt
= 3\sqrt{\cos^2(t)\sin^2(t)}\; dt \;\neq \;3 \cos(t)\sin(t) \;dt
[/tex]

[tex]
3\sqrt{\cos^2(t)\sin^2(t)}\; dt = 3|\cos(t)\sin(t)|\;dt
[/tex]

that's why you have the two contributions cancelling each other out... you have forgotten the absolute value sign...
 

1. What is the moment of inertia of a wire shaped like an astroid?

The moment of inertia (MOI) of a wire shaped like an astroid is a measure of its resistance to changes in rotational motion. It is given by the formula I = (1/10)MR^2, where M is the mass of the wire and R is the radius of the astroid.

2. How do you find the MOI of a wire shaped like an astroid?

To find the MOI of a wire shaped like an astroid, you first need to calculate the mass and radius of the wire. Then, plug these values into the formula I = (1/10)MR^2 to calculate the MOI.

3. What is the significance of finding the MOI of a wire shaped like an astroid?

The MOI of a wire shaped like an astroid is important in understanding its rotational motion. It helps to determine how much torque is required to accelerate or decelerate the wire and how it will behave under different forces.

4. Are there any real-life applications of the MOI of a wire shaped like an astroid?

Yes, the MOI of a wire shaped like an astroid is relevant in various fields such as engineering, physics, and astronomy. It is used in designing structures that need to withstand rotational forces, calculating the stability of satellites and spacecraft, and analyzing the motion of celestial bodies.

5. Can the MOI of a wire shaped like an astroid change?

Yes, the MOI of a wire shaped like an astroid can change if its mass or radius changes. It can also be affected by the distribution of mass along the wire. In addition, external forces such as torque can alter the MOI of the wire, causing changes in its rotational motion.

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