Question

Define constants and initial conditions

m = 30; % kg

c = 0; % Ns/m

k = 38880; % N/m

x0 = 0.1; % m

v0 = 0; % m/s

% Write the differential equation

syms x(t)

eqn = m*diff(x,2) + c*diff(x) + k*x == 0;

cond = [x(0) == x0, diff(x)(0) == v0];

xSol(t) = dsolve(eqn, cond);

% Plot the response

fplot(xSol, [0 20]);

xlabel('Time (s)');

ylabel('Position (m)');

title('Case 1: Undamped System');

Answer 1

The question involves deriving the position function for an undamped simple harmonic oscillator with given constants and initial conditions. The position function is found using energy relationships and is graphed to visualize the oscillatory motion over time, highlighting the system's equations of motion.

Step-by-step explanation:

The student's question involves the mathematical description of a physical system, specifically a simple harmonic oscillator which is characterized by its equations of motion for a block attached to a spring. The constants and initial conditions provided (mass m, damping coefficient c, spring constant k, initial displacement x0, initial velocity v0) define a system without damping owing to a zero coefficient c. The given differential equation represents the second-order linear homogeneous differential equation with constant coefficients describing the motion of the mass.

Using the given energy relationship and initial conditions at t = 0, we can deduce the solution for the displacement x(t) in terms of sinusoidal functions that represent the oscillatory motion of the mass, where the angular frequency is √(k/m) and the initial phase is ±90°. The position function employs a cosine function due to the phase shift. To generate the motion plot, the symbolic solution xSol(t) is graphed over a specified interval of time, thus visualizing the undamped oscillations of the mass-spring system over time.