Final answer:
The equation of motion for a mass attached to a spring can be derived using the second-order differential equation that represents simple harmonic motion, with specific solutions for undamped and damped motions. For undamped systems, the general solution is a cosine function with an amplitude, angular frequency, and phase shift. Damped systems require additional consideration of the damping factor to determine the equation of motion.
Step-by-step explanation:
To find the equation of motion for a mass attached to a spring, we use the second-order differential equation that represents the simple harmonic motion (SHM) of a mass-spring system:
m\(\frac{d^2x}{dt^2}\) + b\(\frac{dx}{dt}\) + kx = 0
In this case:
- m is the mass of the object,
- k is the spring constant,
- x is the displacement from the equilibrium position,
- b is the damping coefficient, and
- \(\frac{dx}{dt}\) is the velocity of the object.
For an undamped and unforced system (b=0), the general solution is:
x(t) = Acos(\(\sqrt{\frac{k}{m}}\)t + \(\phi\))
For a damped and unforced system, the equation of motion depends on the nature of the damping:
- Underdamped: \(\sqrt{k/m} > b/2m\),
- Critically damped: \(\sqrt{k/m} = b/2m\),
- Overdamped: \(\sqrt{k/m} < b/2m\).
If the system is underdamped, the equation of motion is:
x(t) = Ae^{(-b/2m)t}cos(\(\sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}\)t + \(\phi\))
Using the given values for each scenario, apply the relevant formula to find the equations of motion. The amplitude (A), angular frequency (\(\sqrt{k/m}\)), damping factor (b/2m), and phase angle (\(\phi\)) will be determined based on the initial conditions such as displacement and velocity at t=0.