A student pushes a baseball of m=0.19 kg
 down onto the top of a vertical spring that has its lower end fixed to a table, compressing the spring a distance of d=0.18 m
 from its original equilibrium point. The spring constant of the spring is k=630 N/m
. Let the gravitational potential energy be zero at the lowest position of the baseball on the compressed spring.

This is a classic physics problem that involves the concepts of potential energy in both gravitational and elastic forms. Here's how we can approach the problem:

### Key Information:
- Mass of the baseball \( m = 0.19 \, \text{kg} \)
- Compression of the spring \( d = 0.18 \, \text{m} \)
- Spring constant \( k = 630 \, \text{N/m} \)
- Gravitational potential energy is set to zero at the lowest position of the baseball.

### Step 1: Elastic Potential Energy of the Compressed Spring

The potential energy stored in a compressed spring is given by the elastic potential energy formula:
\[
U_{\text{spring}} = \frac{1}{2} k d^2
\]
Substituting the values:
\[
U_{\text{spring}} = \frac{1}{2} \times 630 \, \text{N/m} \times (0.18 \, \text{m})^2
\]

\[
U_{\text{spring}} = \frac{1}{2} \times 630 \times 0.0324
\]
\[
U_{\text{spring}} = 10.21 \, \text{J}
\]

So, the potential energy stored in the spring is **10.21 J**.

### Step 2: Gravitational Potential Energy
Since the gravitational potential energy is set to zero at the lowest point (where the spring is compressed), the gravitational potential energy at this position is:
\[
U_{\text{gravity}} = 0 \, \text{J}
\]

At any higher point, gravitational potential energy can be calculated using:
\[
U_{\text{gravity}} = mgh
\]
where \( h \) is the height above the lowest point.

### Step 3: Mechanical Energy Conservation

When the baseball is released, assuming no energy is lost (no friction), the total mechanical energy is conserved. The total energy is the sum of gravitational and spring potential energies. At the point when the baseball leaves the spring, all the spring's potential energy will have converted to kinetic energy and gravitational potential energy.

If you need more information, feel free to ask!

### Would you like more details or have any other questions? 

Here are 5 follow-up questions to expand on this topic:

1. What would be the velocity of the baseball at the moment it leaves the spring?
2. How does the gravitational potential energy change as the baseball moves upwards?
3. What is the maximum height the baseball reaches after being released from the spring?
4. How does the energy stored in the spring compare to the work done on the baseball?
5. How would the situation change if the spring constant were halved?

**Tip:** In problems involving energy conservation, it's helpful to identify points where potential energy (gravitational or elastic) is maximum or minimum, as these often simplify calculations.

A student pushes a baseball of m=0.19 kg down onto the top of a vertical spring that has its lower end fixed to a table, compressing the spring a distance of d=0.18 m from its original equilibrium point. The spring constant of the spring is k=630 N/m. Let the gravitational potential energy be zero at the lowest position of the baseball on the compressed spring.

Explore the energy conservation principles behind compressing a spring with a baseball weighing 0.19 kg and a spring constant of 630 N/m. Learn how to calculate the elastic potential energy and understand mechanical energy conservation. Suitable for Grades 11-12.

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