A 0.180 kg block on a vertical spring with spring constant of 4.00 ✕ 103 N/m is pushed downward, compressing the spring 0.061 m. When released, the block leaves the spring and travels upward vertically. How high does it rise above the point of release?
To solve this problem, we can use the principle of conservation of mechanical energy. When the block is released, the energy stored in the compressed spring is converted into the potential energy of the block as it rises. So, we need to find the potential energy of the block at the highest point.
First, let's find the potential energy stored in the compressed spring. The potential energy stored in a spring is given by the formula:
PEspring = (1/2)kx^2
Where PEspring is the potential energy stored in the spring, k is the spring constant, and x is the displacement of the spring from its equilibrium position.
In this problem, the spring constant is given as 4.00 ✕ 10^3 N/m and the displacement is given as 0.061 m. Substituting these values into the formula, we get:
PEspring = (1/2)(4.00 ✕ 10^3 N/m)(0.061 m)^2
= 4.00 ✕ 10^3 N/m ✕ 0.001861 m^2
= 7.44 J
Next, let's find the potential energy of the block at the highest point. The potential energy of an object near the Earth's surface is given by the formula:
PEgravity = mgh
Where PEgravity is the potential energy due to gravity, m is the mass of the object, g is the acceleration due to gravity, and h is the height above a reference point.
In this problem, the mass of the block is given as 0.180 kg. We can assume that the reference point is the point of release. Also, the block rises vertically, so the displacement h is equal to the initial displacement of the spring, which is 0.061 m. Substituting these values into the formula, we get:
PEgravity = (0.180 kg)(9.8 m/s^2)(0.061 m)
= 0.107 J
Finally, to find the height above the point of release, we need to equate the potential energy due to gravity to the potential energy stored in the spring:
PEgravity = PEspring
0.107 J = 7.44 J
Since the potential energy values don't match, it means that the block does not rise above the point of release. It reaches equilibrium and then falls back down.
Therefore, the height above the point of release is 0 meters.