An object of mass = 4.00 is attached to a spring with spring constant = 176 whos

Question

An object of mass = 4.00 is attached to a spring with spring constant = 176 whose unstretched length is = 0.120 , and whose far end is fixed to a shaft that is rotating with an angular speed of = 2.00 . Neglect gravity and assume that the mass also rotates with an angular speed of 2.00 as shown. (Intro 1 figure) When solving this problem use an inertial coordinate system, as drawn here. (Intro 2 figure)


Given the angular speed of = 2.00 , find the radius at which the mass rotates without moving toward or away from the origin.
Express your answer in meters.
=

Explanation / Answer

Equate the centripetal force Fc with the tension force in the spring Ft: Let R be R(omega). Then Fc = M * w^2 * R Ft = k (R – L ) (1) M * w^2 * R = k (R – L ) Solving for R: (2) M * w^2 * R = k * R – k * L (3) - M * w^2 * R + k * R = k * L (4) ( - M * w^2 + k) * R = k * L (5) (k - M * w^2) * R = k * L (6) R = k * L / (k - M * w^2) and (7) R = L / (1 - M * w^2 / k) Substituting the values below: k = 1380 N/m L = 0.150 m M = 5 Kg w = 5 rad/s (8) R = 0.150 / (1 - 5 * 5^2 / 1380) = 0.165 m

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