A particle undergoes damped harmonic motion.The spring constant is 100 N/m; the damping constant is 8.0 x 10^-3 kg∙m/s, and the mass is 0.050 kg.If the particle starts at its maximum displacement, x = 1.5 m, at time t = 0, what is the particle's position at t = 5.0 s? A)-1.5 m B)-0.73 m C)0 m D)0.73 m E)1.5 m

Question

Quizplus · Accepted Answer

First, we find the angular frequency of the system: 
ω = sqrt(k/m) = sqrt(100/0.050) = 20 rad/s

Next, we find the damping ratio: 
ζ = c/(2mω) = (8.0 x 10^-3)/(2*0.050*20) = 0.02

Since ζ is less than 1, this is an underdamped system. 
The position of the particle at time t can be given by: x = A*e^(-ζωt)*cos(ωsqrt(1-ζ^2)*t + φ)

At t = 0, x = A*cos(φ) = 1.5 m (given in the problem). 
Also, the velocity of the particle at t = 0 is 0.

Using these initial conditions, we can solve for A and φ: 
A = 1.5 m
φ = 0 radians

Therefore, the position of the particle at t = 5.0 s is: 
x = 1.5*e^(-0.02*20*5)*cos(20*sqrt(1-0.02^2)*5) 
x = -0.73 m

Therefore, the answer is (B).

A Particle Undergoes Damped Harmonic Motion