Question: Three identical thin rods, each of length L and mass m, are welded perpendicular to one another as shown in Figure P10.43. The assembly is rotated about an axis that passes through the end of one rod and is parallel to another. Determine the moment of inertia of this structure about this axis.
The moment of inertia of this structure about this axis
Moment of inertia about a given axis of rotation resists a change in its rotational motion; it can be regarded as a measure of rotational inertia of the body.
This problem is solved by calculating the moment of inertia of the three rods about the y-axis. Then, make use of the fact that the rotation axis is parallel to the y-axis a distance of away and apply parallel axis theorem to determine the moment of inertia about the given axis. The diagram is shown below.
The moment of inertia of a long, thin rod with rotation axis is
The moment of inertia of the rod lying in the x-axis is
Similarly, the moment of inertia of the rod lying in the z-axis is
And the rod lying in the y-axis contributes zero moment about its own axis.
Hence, the total moment of inertia about y-axis which lies at the center of mass of the rod that lies in the x and z axes
Then, according to parallel-axis theorem, the moment of inertia of the system of the three rods about the y-axis is
where (M) is the total mass of the three rods which are identical.
Thus, the solution is .
Question: The fishing pole in Figure P10.28 makes an angle of with the horizontal. What is the torque exerted by the fish about an axis perpendicular to the page and passing through the angler’s hand if the fish pulls on the fishing line with a force at an angle below the horizontal? The force is applied at a point from the angler’s hands.
Question: A wheel 2.00m in diameter lies in a vertical plane and rotates about its central axis with a constant angular acceleration of . The wheel starts at rest at t=0, and the radius vector of a certain point on the rim makes an angle of with the horizontal at this time. At, find (a) the angular speed of the wheel and, for point P, (b) the tangential speed, (c) the total acceleration, and (d) the angular position.
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