A 275g ball is resting on top of the spring that is mounted on the floor. You exert a force of 325N on the ball and it compresses the spring 44.5cm. If you release the ball from the position, how high above the equilibrium position of the spring-ball system will the ball rise?

Thats an easy one

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To find how high the ball will rise above the equilibrium position of the spring-ball system, we can apply the principle of conservation of mechanical energy.

First, let's calculate the potential energy stored in the compressed spring when the ball is at the maximum compression. The potential energy stored in a spring is given by the formula:

Potential Energy = (1/2) * k * x^2

Where:
k is the spring constant (measured in N/m)
x is the compression or extension of the spring (measured in meters)

We are given that the spring is compressed by 44.5 cm, which is 0.445 meters.
We can find the spring constant (k) by using Hooke's Law, which states that the restoring force of a spring is directly proportional to the displacement from the equilibrium position. Mathematically, it is represented as F = -kx, where F is the force applied, k is the spring constant, and x is the displacement.

In our case, we know that the restoring force exerted by the spring is 325 N when the compression is 0.445 m. So, we can write the equation as:

325 N = -k * 0.445 m

Now, we can solve for the spring constant (k):

k = -325 N / 0.445 m
k ≈ -730 N/m (Note: We take the negative sign because the spring exerts a restoring force in the opposite direction of the compression)

Now, we can calculate the potential energy stored in the compressed spring:

Potential Energy = (1/2) * k * x^2
Potential Energy = (1/2) * (-730 N/m) * (0.445 m)^2
Potential Energy ≈ 71.23125 J (rounded to four decimal places)

The potential energy stored in the compressed spring is approximately 71.23125 J.

According to the principle of conservation of mechanical energy, this potential energy will be converted into kinetic energy as the ball rises above the equilibrium position of the spring-ball system.

At the highest point, all the potential energy is converted into kinetic energy, so we can write the equation:

Potential Energy = Kinetic Energy

m * g * h = (1/2) * m * v^2

Where:
m is the mass of the ball (275 g or 0.275 kg)
g is the acceleration due to gravity (9.8 m/s^2)
h is the maximum height above the equilibrium position
v is the velocity of the ball when it reaches the highest point (which is 0 m/s)

We know that the mass (m) cancels out in this case, so:

g * h = (1/2) * v^2
h = (1/2) * v^2 / g

Since the ball comes to rest at the highest point, the velocity (v) is 0. So we have:

h = (1/2) * 0^2 / g = 0

Therefore, the ball does not rise above the equilibrium position of the spring-ball system.