# Moment of Inertia of 'U' shaped rod hanging from celling [closed]

The ends of three identical uniform thin rods are joined at right angles to form a U-shaped body. A freely rotating axle $A$ is attached to one end of this body. The axle is then fixed to the ceiling such that the body hangs freely from one end. Assume that the system is at equilibrium.

Part A) Draw a free-body diagram of the U-shaped body, and deduce the angle made by its upper leg with the vertical.

Part B) Let each thin rod in the U-shaped body have mass $m$ and length $l$. Find the moment of inertia of the body about the axle $A$. Assume that the axis of rotation is perpendicular to the plane defined by the body

Working -

With part A, when the U-shaped rod is in equilibrium, I suspected that the centre of mass of the rod would move such that it would be along the perpendicular of the axle $A$. From there I really am unsure on how to determine the angle it will make. With Part B, I do know that I can use the Perpendicular axis theorem but that's where I get stuck as well.

Any suggestions, directions, advice or even general information about how to tackle these problems would be greatly appreciated. Feel free to edit my question for clarity. Thank you

## closed as off-topic by Gert, user36790, ACuriousMind♦, John Rennie, Kyle KanosNov 17 '15 at 11:16

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The key in either case will be the parallel axis theorem: $I = I_{cm} + m d^2$ where $m$ is the mass of a single rod and d is the distance from the center-of-mass axis to a parallel axis. Note that the center-of-mass axis is referring to an axis through the center of mass of a single rod (not the combined center of mass mentioned above). In other words, d is the distance from point A to the center of a particular thin rod.