A 3.60 kg block starts from rest and slides down a frictionless incline, dropping a vertical distance of 2.80 m, before compressing a spring of force constant 2.30*10^4 N/m. Find the maximum compression of the spring.

Any help would be appreciated, this problem is to use Spring Potential Energy; Conservation of Energy. If you could explain how you get the answer it would be greatly appreciated.

To find the maximum compression of the spring, we can use the principle of conservation of energy, which states that the total mechanical energy of a system remains constant if only conservative forces are acting on it.

In this scenario, the block starts from rest, so its initial kinetic energy is zero. The only forces acting on the block are the gravitational force and the force of the spring.

The potential energy due to gravity at the top of the incline is given by the formula: PEgravity = m * g * h
where m is the mass of the block, g is the acceleration due to gravity, and h is the vertical distance dropped.

The potential energy stored in the compressed spring when maximally compressed is given by the formula: PEspring = (1/2) * k * x^2
where k is the force constant of the spring, and x is the compression of the spring.

Since there is no friction, the mechanical energy at the top of the incline (potential energy due to gravity) is equal to the mechanical energy at the maximum compression of the spring (potential energy stored in the spring). Therefore, we can equate these energies:

PEgravity = PEspring

m * g * h = (1/2) * k * x^2

Now we can plug in the given values:

m = 3.60 kg (mass of the block)
g = 9.8 m/s^2 (acceleration due to gravity)
h = 2.80 m (vertical distance dropped)
k = 2.30 * 10^4 N/m (force constant of the spring)

Substituting these values into the equation:

3.60 kg * 9.8 m/s^2 * 2.80 m = (1/2) * (2.30 * 10^4 N/m) * x^2

Simplifying the equation:

99.144 = 11500 * x^2

Dividing both sides by 11500:

0.008618 = x^2

Taking the square root of both sides:

x = 0.093 m

Therefore, the maximum compression of the spring is 0.093 meters or 9.3 cm.