Figure 1 An object of mass M M, and ? L , M k, L R (omega) at which the mass rot

Question

Figure 1

An object of mass M M, and ? L , M k, L R (omega) at which the mass rotates without moving toward or away from the origin. Express the radius in terms of k omega, find the radius R(?) Given the angular speed ? omega as shown. (Figure 1) When solving this problem use an inertial coordinate system, as drawn here. (Figure 2) Figure 1 Figure 2 omega. Neglect gravity and assume that the mass rotates with angular speed ? L, and whose far end is fixed to a shaft that is rotating with angular speed ? K whose unstretched length is L M is attached to a spring with spring constant k

Explanation / Answer

You need to find the radius where the force on the mass gives the centripetal acceleration that is needed to move in a circle without changing the radius.

Centripetal accel = r ?^2
Force in spring = k(r-L) whre L is the unstretched length
So r ?^2 = k(r-L)/m
m r ?^2 = kr - kL
(m ?^2 - k)r = -kL
r = kL / (k - m ?^2)

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