A 0.230 kg block on a vertical spring with spring constant of 4.25e3 N/m is pushed downward, compressing the spring 0.048 m. When released, the block leaves the spring and travels upward vertically. How high does it rise above the point of release?

To determine how high the block rises above the point of release, we can use the conservation of mechanical energy principle. This principle states that the total mechanical energy (sum of potential and kinetic energy) of a system remains constant as long as there are no external forces acting on it.

In this case, the initial potential energy of the block is stored in the compressed spring, and when released, this potential energy is converted into kinetic energy as the block travels upward.

First, let's find the potential energy stored in the spring when it is compressed. The potential energy of a spring is given by the equation:

Potential energy (U) = (1/2) * k * x^2

Where k is the spring constant and x is the displacement from the equilibrium position (in this case, the compression of the spring).

Substituting the given values:

Potential energy (U) = (1/2) * (4.25e3 N/m) * (0.048 m)^2

Now, let's calculate the potential energy (U) and convert it into kinetic energy (K) when the block reaches its maximum height. At this point, all of the initial potential energy is converted into kinetic energy.

Potential energy (U) = 0 (at maximum height)
Kinetic energy (K) = U

Using the principle of conservation of mechanical energy:

Potential energy (U) + Kinetic energy (K) = Total mechanical energy

0 + U = Total mechanical energy

Now, equating the potential and kinetic energies:

(1/2) * (4.25e3 N/m) * (0.048 m)^2 = (1/2) * m * v^2

Where m is the mass of the block and v is its velocity at the maximum height.

Next, let's calculate the velocity of the block at the maximum height. We can use the principle of conservation of mechanical energy again:

Total mechanical energy = Potential energy (at release) + Kinetic energy (at maximum height)

Total mechanical energy = (1/2) * k * x^2 + (1/2) * m * v^2

Substituting the given values:

Total mechanical energy = (1/2) * (4.25e3 N/m) * (0.048 m)^2 + (1/2) * (0.230 kg) * v^2

Now, we can solve for v, the velocity at the maximum height.

Next, we can use the equation of motion to find the height:

v^2 = u^2 - 2 * g * h

Where u is the initial velocity (which is zero in this case), g is the acceleration due to gravity (9.8 m/s^2), and h is the height.

Substituting the values:

v^2 = 0 - 2 * (9.8 m/s^2) * h

Now, plug in the previously calculated value of v and solve for h, which is the height the block rises above the point of release.