- Derive equation (2) using the hints given in the theory section of this handout.
Derive Equation 2 Using The Hints Given In The Theory Section Of This Handout 1 (81.24 KiB) Viewed 12 times
Derive Equation 2 Using The Hints Given In The Theory Section Of This Handout 2 (129.16 KiB) Viewed 12 times
Theory The dynamic analysis of the rotational motion apparatus yields the following equation W= mgr - Tf (mgr (1) I- t (1) where w is the angular velocity of the aluminum disk in [rad/s], m is the hanging mass in [g], g = 981 cm/s² is the gravitational acceleration constant, r is the pulley radius in [cm], Tf is the frictional torque in [dyne cm], I is the combined moment of inertia of the aluminum disk and the step pulley in [g-cm²], and t is the time in [s].
When the experiment is conducted for two different hanging mass values mi and m2, the angular velocity @ vs time t data yields two different slope values, S₁ and S₂. Using the part of equation (1) that represents the slope, two additional equations may be created which are equal to S₁ and S₂. Using these two equations, the frictional torque tf is eliminated and an equation that provides a solution for the combined moment of inertia of the aluminum disk and the step pulley I is generated as shown below. Tf I = (m₂ − m₁)gr + (S₁m₁ − S₂m₂)r² S₂ - S₁ (2) where my and m2 are the hanging masses 1 and 2 in [g], g = 981 cm/s² is the gravitational constant, S, and S2 are the two slope values that will be determined from the experimental measurements, and r is the pulley radius in [cm].
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