Damped Oscillations: Mass 300 g, k=1.50 N/m, b in kg/s

[physicsgurl123]

A weight of mass m = 300 g hangs vertically from a spring that has a spring constant k = 1.50 N/m. The mass is set into vertical oscillation and after 28 s you find that the amplitude of the oscillation is 1/10 that of the initial amplitude. What is the damping constant b associated with the motion (in kg/s)? Also what would the graphs of the mechanical and kinetic energies look like?

 

[Andrew Mason]

A weight of mass m = 300 g hangs vertically from a spring that has a spring constant k = 1.50 N/m. The mass is set into vertical oscillation and after 28 s you find that the amplitude of the oscillation is 1/10 that of the initial amplitude. What is the damping constant b associated with the motion (in kg/s)? Also what would the graphs of the mechanical and kinetic energies look like?

What is the equation of motion here? Hint: The general form of solution is:

$$x = A_0e^{-qt}\sin(\omega t + \phi)$$AM

 

[kyle8921]

Based on the given information, we can use the equation for damped oscillations to determine the damping constant b:

mω = b/2m

Where m is the mass of the weight (300 g) and ω is the angular frequency, which can be calculated using the formula:

ω = √(k/m)

Substituting the values, we get:

ω = √(1.50 N/m / 0.3 kg) = √5 rad/s

Now, using the given time and amplitude information, we can calculate the new angular frequency ω' using the formula:

ω' = ω / (1/10)

ω' = 10ω = 50 rad/s

Substituting this value into the equation for damping constant, we get:

b = 2mω' = 2 x 0.3 kg x 50 rad/s = 30 kg/s

Therefore, the damping constant b associated with the motion is 30 kg/s.

As for the graphs of mechanical and kinetic energies, the mechanical energy graph would show a decrease in amplitude over time, as the energy is being dissipated due to damping. The kinetic energy graph would also show a decrease in amplitude, but at a faster rate compared to the mechanical energy graph, as the kinetic energy is directly proportional to the square of the amplitude. Eventually, both graphs would reach a steady state where the energy remains constant, indicating that the oscillations have stopped due to the damping force balancing out the restoring force of the spring.