A thin rod of mass M, length L with the left end at the origin has a density function λ= Ao [1 - (x²/ L²)] and rest on the positive x-axis. Relate M with density constant λo. Find the moment of inertia in terms of M and L only. Find the center of mass in terms of L only. Find the angular velocity w₂ if this rod swings to vertical (pivoted at the origin). The bottom edge of the rod (now vertically hanging) is attached to a spring of constant k and set up as an oscillator. The spring is unstretched when the rod is vertical. Relate the torque on the rod with the angular acceleration. Then find the angular frequency of the oscillation.

College Physics
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Chapter1: Units, Trigonometry. And Vectors
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A thin rod of mass M, length L with the left end at the
origin has a density function λ= Ao [1 − (x²/ L²)] and rest
on the positive x-axis. Relate M with density constant o.
Find the moment of inertia in terms of M and L only. Find
the center of mass in terms of L only. Find the angular
velocity w₂ if this rod swings to vertical (pivoted at the
origin). The bottom edge of the rod (now vertically
hanging) is attached to a spring of constant k and set up
as an oscillator. The spring is unstretched when the rod is
vertical. Relate the torque on the rod with the angular
acceleration. Then find the angular frequency of the
oscillation.
Transcribed Image Text:A thin rod of mass M, length L with the left end at the origin has a density function λ= Ao [1 − (x²/ L²)] and rest on the positive x-axis. Relate M with density constant o. Find the moment of inertia in terms of M and L only. Find the center of mass in terms of L only. Find the angular velocity w₂ if this rod swings to vertical (pivoted at the origin). The bottom edge of the rod (now vertically hanging) is attached to a spring of constant k and set up as an oscillator. The spring is unstretched when the rod is vertical. Relate the torque on the rod with the angular acceleration. Then find the angular frequency of the oscillation.
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