A student has a spring, whose spring constant is 5.0 N/m, with a 1.0-kg mass attached to it that is hanging freely. The quality factor of the system is 50. She wishes to have the maximum displacement of the system be 7.5 cm. The force and frequency required to produce this motion are
A) 0.0075 N and 2.8 s.
B) 0.0075 N and 1.4 s.
C) 19 N and 2.8 s.
D) 19 N and 1.4 s.

Question

Quizplus · Accepted Answer

The maximum displacement of a damped harmonic oscillator is given by:$$x_{max} = \frac{F_0}{k} \frac{1}{\sqrt{1+(2\zeta)^2}}$$where $F_0$ is the driving force, $k$ is the spring constant, $\zeta$ is the damping ratio, and $x_{max}$ is the maximum displacement.We are given that $x_{max} = 7.5 cm = 0.075 m$, $k = 5.0 N/m$, and $\zeta = 1/Q = 1/50 = 0.02$. Substituting these values into the equation, we get:$$0.075 m = \frac{F_0}{5.0 N/m} \frac{1}{\sqrt{1+(2	imes 0.02)^2}}$$Solving for $F_0$, we get:$$F_0 = 0.0075 N$$The frequency of the driving force is given by:$$f = \frac{1}{2\pi} \sqrt{\frac{k}{m} - \zeta^2}$$Substituting the given values, we get:$$f = \frac{1}{2\pi} \sqrt{\frac{5.0 N/m}{1.0 kg} - (0.02)^2} = 2.8 s^{-1}$$Therefore, the correct answer is A) 0.0075 N and 2.8 s.

A Student Has a Spring, Whose Spring Constant Is 5