Solving Frequency of Damped Oscillations

[pilotman]

Homework Statement

A mass of 0.5kg hangs on a spring. When an additional mass of 0.2kg is attached to the spring, the spring stretches an additional .04m. When the 02kg mass is abruptly removed, the amplitude of the ensuing oscillations of the 05 kg mass is observed to decrease to 1/e of its initial value in 1.0s. Calculate the frequency of the damped oscillations.

Homework Equations

The Attempt at a Solution

I'm not sure how to start this problem. What I'm looking for is someone to give me a push in the right direction so that I can work the rest! please help me start! Thanks all!

 

[anathema]

Thank you for your question. It seems like you are dealing with a damped harmonic oscillator in this problem. To start, you will need to use the equation for the period of a damped harmonic oscillator, which is given by T=2π/ω, where T is the period and ω is the angular frequency. To find the angular frequency, you will need to use the equation ω=√(k/m), where k is the spring constant and m is the mass. You can find the spring constant by using Hooke's Law, which states that F=-kx, where F is the force, k is the spring constant, and x is the displacement from equilibrium. Once you have the angular frequency, you can use the equation for the period to find the frequency, which is given by f=1/T. I hope this helps you get started on solving the problem. Good luck!

 

[nlightner]

I would approach this problem by first identifying the key variables and their relationships. In this case, we have a mass (m), a spring constant (k), and a damping factor (e). These variables are related through the equation for damped oscillations:

f = 1/2π * √(k/m - (e/2m)^2)

To solve for the frequency, we need to first find the values for k and e. We can do this by using the given information about the masses and the displacement of the spring.

From the given information, we can calculate the spring constant (k) using the equation F = kx, where F is the force exerted by the spring and x is the displacement.

F = mg = (0.5kg)(9.8m/s^2) = 4.9N

Using this value for F and the given displacement (.04m), we can solve for k:

k = F/x = (4.9N)/(.04m) = 122.5 N/m

Next, we can use the information about the damping factor (e) to solve for its value. We know that the amplitude decreases to 1/e of its initial value in 1.0s, so we can set up the equation:

1/e = A/A0 = e^(-1)

Solving for e, we get e = 1/e^(-1) = 0.3678

Now, we have all the necessary values to plug into the equation for frequency:

f = 1/2π * √(122.5 N/m / (0.5kg) - (0.3678/2(0.5kg))^2)

Simplifying, we get f ≈ 0.79 Hz

Therefore, the frequency of the damped oscillations is approximately 0.79 Hz.