I am attempting to verify the solution to a variable angular inertia problem. Initially, the problem is set up this way:(adsbygoogle = window.adsbygoogle || []).push({});

Given a sphere with a diameter of 0.125 meter and a massms=0.997 kg on a massless rod with a perpendicular axle such that the sphere can rotate about the axle at a distance from the center of rotation,_{fw}r, of 15 inches and can increase continuously to 20 inches. The period to increase is 0 to 10 seconds, t. Thus,ras a function oftis:r(t)=15…20 inches.

The moment of inertia is calculated using the parallel-axis theorem (Steiner’s Theorem):

Iwhere_{vr}(t) = 2/5*ms_{fw}*R^{2}+ ms_{fw}*r(t)^{2}Ris the radius of the sphere

The angular velocity is [itex]\omega[/itex] and is to remain constant at 1000 rpm. The question to answer is: What is the torque([itex]\tau[/itex])input on the axle required to maintain a constant angular velocity ([itex]\omega[/itex])?

Since torque equalsdL/dtthe solution is simple:[itex]\tau[/itex](t) =[ (I_{vr}(t_{i})*_{w})-(I_{vr}(t_{e})*[itex]\omega[/itex])]/t = 1.226 N*m

To check this answer, I attempted to solve the same physical set up using linear momentum. Treating the sphere as a point-mass, I calculated the linear velocity based on the angular velocity (1000 rpm). Then calculating the linear velocity of the mass at 15 in. radius and also at 20 inch radius, I then used the two different velocities to calculate the beginning momentum and the ending momentum. Then using dp/dt to find the resultant force on the mass and lever arm (rod), torque equals force timesr.

The result I get is .703 N*m vs. 1.226 N*m using angular momentum. Can anyone offer an explanation why this is so?

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# Angular momentum dilemma

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