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An object of mass M = 1
An object of mass M = 1.00 kg is attached to a spring with spring constant k = 99.0 N/m whose unstretched length is L = 0.140 m , and whose far end is fixed to a shaft that is rotating with an angular speed of ω = 3.00 radians/s . Neglect gravity and assume that the mass also rotates with an angular speed of 3.00 radians/s as shown. (Figure 1) When solving this problem use an inertial coordinate system, as drawn here. (Figure 2)
Assume that, at a certain angular speed ω2, the radius R becomes twice L. Find ω2.
Express your answer in radians per second.
Expert Solution
x = stretch in the spring = R - L
spring force = centripetal force
kx = m Rw2
99 (R - 0.14) = 1 (R) (3)2
R = 0.154 m
R = 2 L = 2 x 0.14 = 0.28 m
x = stretch in the spring = R - L = 2L - L = L = 0.14 m
M = mass of object = 1 kg
k = spring constant = 99 N/m
spring force = centripetal force
kx = m Rw22
99 (0.14) = 1 (0.28) w22
w2 = 7.04 rad/s
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