Question

Which equations to use and what each value represents in each.

A 2.92kg mass is suspended by a spring with a spring constant of 0.514N/m, and the other end of the spring is affixed to the underside of a wooden shelf which, in turn, is firmly attached to the wall. The mass is given a small vertical displacement, and the system is allowed to oscillate. Electronic data collection apparatus is used to measure the position of the mass, and a graph of position versus time is created. The graph indicates that the system is lightly damped.

A. Students count the number of oscillations that occur until the amplitude has decreased to 63.5% of its initial value. From this, it is determined that the damping coefficient is b = 1.93*10^-3kg/s. How many oscillations did the student count?

B. Students count 169 oscillations that occur until the amplitude has decreased to 75.5% of its initial value. What is the damping coefficient, in kilograms per second, of the system?

Answer 1

Use specific formulas relating to damped SHM to find the number of oscillations and the damping coefficient by working with the equations that involve the damping coefficient, mass, and amplitude decay over time.

Step-by-step explanation:

To solve problems A and B involved with a damped mass-spring system in simple harmonic motion (SHM), we need to use the equation that relates the amplitude decay of a damped oscillator to the damping coefficient and the number of oscillations. The decay of amplitude is given by the equation:

A=A_0 e^{-(b/2m)nt}

where:

• A is the final amplitude
• A_0 is the initial amplitude
• b is the damping coefficient
• m is the mass
• n is the number of oscillations
• t is the period of oscillation

To find the number of oscillations for part A, rearrange the equation to solve for n, and then plug in the values given:

n = (ln(A/A_0) / (-b/2m)) / t

For part B, to find the damping coefficient, we rearrange the equation to solve for b and then use the known values including the number of oscillations:

b = -2m * (ln(A/A_0) / (nt))

These formulas let us analyze the impact of damping on the oscillator's amplitude over time.