To analyze the motion of a harmonic oscillator with damping in a uniform gravitational field, we need to consider the forces acting on the oscillator.

The forces acting on the oscillator can be broken down into three components:
1. The force due to the spring, which is proportional to the displacement from the equilibrium position, given by Hooke's Law.
2. The force due to damping, which is proportional to the velocity of the oscillator and opposes its motion.
3. The force due to gravity, which is proportional to the mass of the oscillator and the acceleration due to gravity.

The equation of motion for the oscillator can be expressed as:

m * d^2x/dt^2 + c * dx/dt + k * x = m * g

Where:
m is the mass of the oscillator,
x is the displacement from the equilibrium position,
t is time,
c is the damping coefficient,
k is the spring constant,
g is the acceleration due to gravity.

To find the equilibrium position of the oscillator, we can set the left-hand side of the equation equal to zero:

0 + 0 + k * x_eq = m * g

This simplifies to:

k * x_eq = m * g

Therefore, the equilibrium position (x_eq) of the oscillator is given by:

x_eq = (m * g) / k

To calculate the oscillation frequency of the oscillator, we can consider the case when the damping coefficient (c) is small compared to the spring constant (k). In this case, we can neglect the damping term in the equation of motion.

The equation becomes:

m * d^2x/dt^2 + k * x = m * g

This is the equation of a simple harmonic oscillator. The angular frequency (ω) of the oscillator, and hence the oscillation frequency (f), can be calculated using the relation:

ω = sqrt(k / m)

f = ω / (2π)

Overall, the equilibrium position of the oscillator is given by (m * g) / k, and the oscillation frequency is given by sqrt(k / m) / (2π).

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