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An object (with mass m) is attached with two identic springs (with spring constant k) to the edge and the axis of a rotating disc (with radius r). The object undergoes no friction and is in the middle between edge and axis. The angular velocity is $\omega$. What is the equilibrium position x in terms of $\omega$, m, k and r?

I have no idea how I'm supposed to start this problem.

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2 Answers 2

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I won't give you the answer but to get you started, use Hooke's law and the centrifugal force expression in terms of the angular velocity ω.

Remember:

  • The centrifugal force depends on the distance of the mass from the axis, x

  • The restoring forces provided by each of the springs are actually the same as each other since one is compressed and the other extended by the same amount (this displacement = x - r/2).

Add the two restoring forces (or multiply one of them by 2), set it equal to (-1 times) the centrifugal force and then solve for x.

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Since the mass is at equilibrium, the force due to rotation which depends on $m,\omega,r$ acting in the radial direction should be equal to the force due to the spring which depends on $k$ and $x$. The springs can be thought to be composed of 2 springs in parallel. Equate them to find $x$

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