Tripod Moment

Classical Mechanics Level 2

A tripod is made of wire with a linear mass density of 1 1 . The tripod consists of four parts:

1) A circular base of radius 1 1 , contained within the x y xy plane, with its center at the origin
2) A straight line segment from ( 1 , 0 , 0 ) (1,0,0) to ( 0 , 0 , H ) (0,0,H)
3) A straight line segment from ( − 1 2 , 3 2 , 0 ) \Big ( -\frac{1}{2}, \frac{\sqrt{3}}{2}, 0 \Big ) to ( 0 , 0 , H ) (0,0,H)
4) A straight line segment from ( − 1 2 , − 3 2 , 0 ) \Big ( -\frac{1}{2}, -\frac{\sqrt{3}}{2}, 0 \Big ) to ( 0 , 0 , H ) (0,0,H)

The attached image shows the projection onto the x y xy plane (to-scale), and the projection onto the y z yz plane (not to-scale).

If the moment of inertia of the entire tripod with respect to the z z axis is 1.5 1.5 times as large as that of the circular base alone, what is the tripod height H H ?


The answer is 2.9782.

Steven Chase by Steven Chase , 37, USA

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1 solution

Moment of inertia of each straight segment about the z-axis is (√(1+H^2))/3, and of the circle is 2π. Therefore by the given condition of the problem, π=√(1+H^2), or H=√(π^2-1)=2.9781881070694

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