MHB Natural frequency of a crane dropping a car

AI Thread Summary
The discussion focuses on calculating the natural frequency of a crane's electromagnet after dropping a car. The natural frequency is determined using the mass of the electromagnet and the car, specifically considering the mass right after release. The motion of the electromagnet is modeled as a simple harmonic oscillator, with the equation reflecting its starting position. Maximum tension in the cable occurs when the mass is at rest, aligning with the maximum acceleration criterion. The participant ultimately resolved the calculations independently.
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An electromagnet weighing \(3000\) lb is at rest while holding an automobile of weight \(2000\) lb in a junkyard. The electric current is turned off, and the automobile is dropped. Assuming that the crane and the supporting cable have an equivalent spring constant of \(10000\) lb/in, find the following:
(1)the natural frequency of vibration of the electromagnet, (2)the resulting motion of the electromagnet, and (3)the maximum tension developed in the cable during motion.

For (1), would it be \(\omega_n = \sqrt{\frac{10000}{5000}}\) or \(\omega_n = \sqrt{\frac{10000}{3000}}\)? Do we look at the mass right before release or right after release?
For (2), this is a simple harmonic oscillator so \(x(t) = A\cos(\omega_nt) + B\sin(\omega_nt)\).
Not sure how to do (3)
 
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For (1), you would look at the mass right after release, since this is the mass that is being accelerated by the cable.
For (2), it would be $$x(t)=-B \cos (\omega t)$$, since the mass starts from the lowest point at t=0.
For (3), remember that the maximum force (tension) occurs when the acceleration is maximum.
Can you proceed?
 
Alternatively for (3), the maximum tension will occur when the mass is at rest (this is just a restatement of the maximum acceleration criterion).
 
jacobi said:
For (1), you would look at the mass right after release, since this is the mass that is being accelerated by the cable.
For (2), it would be $$x(t)=-B \cos (\omega t)$$, since the mass starts from the lowest point at t=0.
For (3), remember that the maximum force (tension) occurs when the acceleration is maximum.
Can you proceed?

Thanks but I figured out everything yesterday.
 
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