an object of mass 1.5kg on a spring of force constant 600N/m looses 3% of its energy per each cycle. the same system is driven by a sinusoidal force with the maximum value of .05N and the driven angural frequency is w=19rad/sec. find the time constant
To find the time constant for this system, we need to make use of the information given about the energy loss per cycle.
The time constant, denoted as τ (tau), is a characteristic of the system that determines how quickly it responds to changes. It is typically related to the damping coefficient, which describes the amount of energy dissipated over time.
In this case, we know that the object loses 3% of its energy per cycle. This loss of energy indicates that there is damping present in the system. We can relate the damping coefficient, b, to the energy loss using the following formula:
Energy loss per cycle = (2πb / m) * E
Where:
- Energy loss per cycle is given as 3% of the total energy (E) of the system.
- m is the mass of the object (1.5 kg).
Simplifying the equation, we have:
0.03E = (2πb / 1.5) * E
Now, let's solve for b (the damping coefficient):
0.03 = 2πb / 1.5
b = 0.03 * 1.5 / (2π)
b ≈ 0.002848
The damping coefficient is approximately 0.002848.
Next, we can relate the force constant, k, to the angular frequency, ω (omega):
k = mω^2
Given that k = 600 N/m, we can solve for ω:
600 = 1.5 * ω^2
ω^2 = 400
ω = √400
ω = 20 rad/sec
Now, we can calculate the time constant, τ, using the following formula:
τ = 1 / (b * ω)
Substituting the values we found, we get:
τ = 1 / (0.002848 * 20)
τ ≈ 17.52 seconds
Therefore, the time constant for the given system is approximately 17.52 seconds.