Consider a variation of Atwood's machine with a mass hanging from a string wrapped around a pulley.
The expressions for the height of the mass and balance of forces are the same as before. For the pulley we need to balance the torque, the product of tension and its distance from the center of the pulley, with the inertial force due to the acceleration of the pulley. It is assumed that the quantity of motion for the pulley is proportional to the rate of change of the angle θ or P=Iω=Iθ̇ where I is the moment of inertia. Note that the resistance to angular acceleration is directed upwards. This time the inertial factor, m+I/r2, includes the inertia of the pulley.
And the tension in the string is again less than the gravitational force acting on the mass m. Barton's Analytic Mechanics derives the formulas for Atwood's machine including the inertia of the pulley.
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