Respuesta :
Answer:
(a) I_A=1/12ML²
(b) I_B=1/3ML²
Explanation:
We know that the moment of inertia of a rod of mass M and lenght L about its center is 1/12ML².
(a) If the rod is bent exactly at its center, the distance from every point of the rod to the axis doesn't change. Since the moment of inertia depends on the distance of every mass to this axis, the moment of inertia remains the same. In other words, I_A=1/12ML².
(b) The two ends and the point where the two segments meet form an isorrectangle triangle. So the distance between the ends d can be calculated using the Pythagorean Theorem:
[tex]d=\sqrt{(\frac{1}{2}L) ^{2}+(\frac{1}{2}L) ^{2} } =\sqrt{\frac{1}{2}L^{2} } =\frac{1}{\sqrt{2} } L=\frac{\sqrt{2} }{2} L[/tex]
Next, the point where the two segments meet, the midpoint of the line connecting the two ends of the rod, and an end of the rod form another rectangle triangle, so we can calculate the distance between the two axis x using Pythagorean Theorem again:
[tex]x=\sqrt{(\frac{1}{2}L)^{2}-(\frac{\sqrt{2}}{4}L) ^{2} } =\sqrt{\frac{1}{8} L^{2} } =\frac{1}{2\sqrt{2}} L=\frac{\sqrt{2}}{4} L[/tex]
Finally, using the Parallel Axis Theorem, we calculate I_B:
[tex]I_B=I_A+Mx^{2} \\\\I_B=\frac{1}{12} ML^{2} +\frac{1}{4} ML^{2} =\frac{1}{3} ML^{2}[/tex]
A) Moment of inertia about an axis passing through the point where the two segments meet : [tex]I_{A} = \frac{1}{12} ML^{2}[/tex]
B) Moment of inertia passing through the point where the midpoint of the line connects to its two ends : [tex]Ix_{} = \frac{1}{3} ML^{2}[/tex]
A) The moment of inertia about an axis passing through the point where the two segments meet is [tex]I_{A} = \frac{1}{12} ML^{2}[/tex] given that the rod is bent at the center and distance from all the points to the axis remains the same, the moment of inertia about the center will remain the same.
B) Determine the moment of inertia about an axis passing through the point midpoint of the line which connects the two ends
First step: determine the distance between the ends ( d )
After applying Pythagoras theorem
d = [tex]\frac{\sqrt{2} }{2} L[/tex]
Next step : determine distance between the two axis ( x )
After applying Pythagoras theorem
x = [tex]\frac{\sqrt{2} }{4} L[/tex]
Final step : Calculate the value of Iₓ
applying Parallel Axis Theorem
Iₓ = Iₐ + Mx²
= [tex]\frac{1}{12} ML^{2}[/tex] + [tex]\frac{1}{4} ML^{2}[/tex]
∴ [tex]Ix_{} = \frac{1}{3} ML^{2}[/tex]
Hence we can conclude that Moment of inertia about an axis passing through the point where the two segments meet : [tex]I_{A} = \frac{1}{12} ML^{2}[/tex], Moment of inertia passing through the point where the midpoint of the line connects its two ends : [tex]Ix_{} = \frac{1}{3} ML^{2}[/tex]
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