A 0.250 kg block on a vertical spring with a

spring constant of 4.09 × 10
3
N/m is pushed
downward, compressing the spring 0.0900 m.
When released, the block leaves the spring
and travels upward vertically.
The acceleration of gravity is 9.81 m/s
2
.
How high does it rise above the point of
release?
Answer in units of m

Call height at spring unextended = 0

then height at bottom = -.09
height at top = h
(we want h +.09)
velocity is 0 at bottom and again at top so we are talking all potential energy here

At bottom total Pe = (1/2)k (.09)^2 - m g(.09)

At top total Pe = (1/2) k h^2+ m g h

those are the same so
(1/2)k h^2 + m g h=(1/2)k(.09)^2-.09 m g

solve for h, add .09

To find how high the block rises above the point of release, we can use the principles of conservation of mechanical energy. The potential energy stored in the spring when it is compressed is then converted into gravitational potential energy as the block rises.

1. Determine the potential energy stored in the spring:
The potential energy stored in a spring is given by the formula: PE = (1/2)kx^2, where k is the spring constant and x is the displacement of the spring from its equilibrium position.
In this case, the spring constant (k) is given as 4.09 × 10^3 N/m, and the displacement (x) is given as 0.0900 m.
Using the formula, we can calculate the potential energy stored in the spring: PE = (1/2)(4.09 × 10^3 N/m)(0.0900 m)^2.

2. Determine the maximum height reached by the block:
The potential energy stored in the spring is converted into gravitational potential energy as the block rises vertically. Therefore, at the maximum height, the potential energy stored in the spring is equal to the gravitational potential energy.
The gravitational potential energy (PE_grav) is given by the formula: PE_grav = mgh, where m is the mass of the block and h is the height above the release point.
In this case, the mass (m) is given as 0.250 kg, and the acceleration due to gravity (g) is given as 9.81 m/s^2.
We can equate the potential energy stored in the spring to the gravitational potential energy and solve for h: PE = mgh.

3. Substitute the values and solve for h:
Substituting the given values, we can solve for h:
(1/2)(4.09 × 10^3 N/m)(0.0900 m)^2 = (0.250 kg)(9.81 m/s^2)h.

Now, performing the calculations, we can find the value of h:

PE = (1/2)(4.09 × 10^3 N/m)(0.0900 m)^2
PE = 16.66645 J

(1/2)(4.09 × 10^3 N/m)(0.0900 m)^2 = (0.250 kg)(9.81 m/s^2)h
16.66645 J = (2.4525 kg·m/s^2)h

Solving for h:
h = (16.66645 J) / (2.4525 kg·m/s^2)
h ≈ 6.7938 m

Therefore, the block rises approximately 6.7938 meters above the point of release.