A 0.25 kg mass is horizontally attached to a spring with a stiffness 4 N/m. The damping constant b for the system is 1 N-sec/m. If the mass is displaced 0.5 m to the left and given an initial velocity of 1 m/sec to the left, find the equation of motion. What is the maximum displacement that the mass will attain?

To find the equation of motion for the system, we can start by applying Newton's second law. The equation of motion for a damped harmonic oscillator can be written as follows:

m * a + b * v + k * x = 0

Where:
m is the mass (0.25 kg)
a is the acceleration
b is the damping constant (1 N-sec/m)
v is the velocity
k is the spring stiffness (4 N/m)
x is the displacement

Differentiating both sides of this equation, we get:

m * dv/dt + b * v + k * x = 0

Since the initial displacement (x) and velocity (v) are given, we can substitute these values into the equation. The equation becomes:

0.25 * d²x/dt² + 1 * dx/dt + 4 * x = 0

This is the equation of motion for the system.

To find the maximum displacement that the mass will attain, we can solve this differential equation.

First, let's convert the second-order differential equation into a characteristic equation by assuming the solution is of the form e^(rt).

Substituting this into the equation, we get:

0.25 * r²e^(rt) + r * e^(rt) + 4 * e^(rt)= 0

Dividing both sides of the equation by e^(rt) to cancel out the exponential term, we have:

0.25 * r² + r + 4 = 0

Now, we can solve this quadratic equation for r using the quadratic formula:

r = (-b ± √(b² - 4ac)) / (2a)

Plugging in the values of a = 0.25, b = 1, and c = 4, we get:

r = (-1 ± √(1 - 4 * 0.25 * 4)) / (2 * 0.25)

Simplifying further:

r = (-1 ± √(1 - 4)) / 0.5
r = (-1 ± √(-3)) / 0.5

Since the term under the square root is negative, the quadratic equation has complex roots. This means that the system is overdamped, and the mass will not oscillate. Instead, it will approach equilibrium exponentially without any maximum displacement.

Therefore, the maximum displacement that the mass will attain is 0.