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## Worked example 8.5: Hinged rod

Question: A uniform rod of mass and length rotates about a fixed frictionless pivot located at one of its ends. The rod is released from rest at an angle beneath the horizontal. What is the angular acceleration of the rod immediately after it is released?

Answer: The moment of inertia of a rod of mass and length about an axis, perpendicular to its length, which passes through one of its ends is (see question 8.3). Hence,

The angular equation of motion of the rod is

where is the rod's angular acceleration, and is the net torque exerted on the rod. Now, the only force acting on the rod (whose line of action does not pass through the pivot) is the rod's weight, . This force acts at the centre of mass of the rod, which is situated at the rod's midpoint. The perpendicular distance between the line of action of the weight and the pivot point is simply

Thus, the torque acting on the rod is

It follows that the rod's angular acceleration is written

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Richard Fitzpatrick 2006-02-02