MHB Natural frequency of a pendulum being lowered at 2m/s

[Dustinsfl]

A heavy machine weighing \(9810\) N is being lowered vertically down by a winch at a uniform velocity of \(2\) m/s. The steel cable supporting the machine has a diameter of \(0.01\) m. The winch is suddenly stopped when the steel cable's length is \(20\) m. Find the period and amplitude of the ensuing vibration of the machine.

From the question, we know that \(m\ddot{x} = 9810\) N, the pendulum will have a length of \(20\) m, and then we have the diameter of cable and the speed at which it was lowered. I am not sure if the last two are superfluous.

How does one tackle this type of problem?

 

[I like Serena]

A heavy machine weighing \(9810\) N is being lowered vertically down by a winch at a uniform velocity of \(2\) m/s. The steel cable supporting the machine has a diameter of \(0.01\) m. The winch is suddenly stopped when the steel cable's length is \(20\) m. Find the period and amplitude of the ensuing vibration of the machine.

From the question, we know that \(m\ddot{x} = 9810\) N, the pendulum will have a length of \(20\) m, and then we have the diameter of cable and the speed at which it was lowered. I am not sure if the last two are superfluous.

How does one tackle this type of problem?

Use Young's modulus for steel.

 

[Dustinsfl]

Use Young's modulus for steel.

It says that the young's modulus for steel is 200.
\[
\omega_n = \sqrt{\frac{200A}{mL}}
\]
where \(A\) is the cross sectional area and \(L\) is the length. So L = 20 and \(A = 0.000025\pi\). The mass is \(m = 9810/g \approx 1000\).
\[
\omega_n = 0.000886227
\]
This doesn't seem right.

 

[I like Serena]

Young's modulus for steel is 200 GPa.

 

[bshoor]

Young's modulus for steel is 200 GPa.

A heavy machine weighing \(9810\) N is being lowered vertically down by a winch at a uniform velocity of \(2\) m/s. The steel cable supporting the machine has a diameter of \(0.01\) m. The winch is suddenly stopped when the steel cable's length is \(20\) m. Find the period and amplitude of the ensuing vibration of the machine.

From the question, we know that \(m\ddot{x} = 9810\) N, the pendulum will have a length of \(20\) m, and then we have the diameter of cable and the speed at which it was lowered. I am not sure if the last two are superfluous.

How does one tackle this type of problem?

From 9810 N you can find the mass of the ball by just dividing 9.81, and mas becomes simply 1000 kg. Now you can find Kinetic Energy of the ball. When the winch is suddenly stopped, the cable gets elongated and a potential energy develops which resist the ball. Now Youngs Modulus gives you the force applied on the cable which comes out as

F = (EA/L)*x=kx where E young's modulus (200 GPa), A area of the cable, L length of the cable, and x is the increment in length.
Elastic potential energy = kx*x/2
Equating with KE, you can get x which gives amplitude.
To find the period apply F=kx or T=2*pi*sqrt(m/k)
In my rough calculation period and amplitude comes about 0.19 sec and 0.035 mt. Best of luck.