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We are tasked with calculating the spring constant \(k\) of a spring that is compressed by a ball dropped from a certain height. Let's break down the given data and approach this step by step:

### Given:
- Mass of the ball: \(m = 147 \, \text{g} = 0.147 \, \text{kg}\)
- Height above the spring: \(h = 52.2 \, \text{cm} = 0.522 \, \text{m}\)
- Maximum compression of the spring: \(x = 4.50803 \, \text{cm} = 0.0450803 \, \text{m}\)
- Acceleration due to gravity: \(g = 9.8 \, \text{m/s}^2\)

### Energy Consideration:
When the ball is dropped, its gravitational potential energy is converted into the elastic potential energy of the spring. Thus, at maximum compression, the total mechanical energy of the system remains conserved.

The gravitational potential energy at the height \(h\) is:
\[
E_{\text{potential}} = mgh
\]

The elastic potential energy stored in the spring when it is compressed by \(x\) is:
\[
E_{\text{spring}} = \frac{1}{2} k x^2
\]

### Energy conservation:
At maximum compression of the spring, the initial gravitational potential energy is equal to the elastic potential energy:
\[
mgh = \frac{1}{2} k x^2
\]

### Solving for \(k\):
\[
k = \frac{2 mgh}{x^2}
\]

Substitute the given values:
\[
k = \frac{2 (0.147 \, \text{kg})(9.8 \, \text{m/s}^2)(0.522 \, \text{m})}{(0.0450803 \, \text{m})^2}
\]

Now, calculating each part:

1. \(mgh = 0.147 \times 9.8 \times 0.522 = 0.7527492 \, \text{J}\)
2. \(x^2 = (0.0450803)^2 = 0.002031 \, \text{m}^2\)
3. Thus,
\[
k = \frac{2 \times 0.7527492}{0.002031} \approx 741.51 \, \text{N/m}
\]

### Final Answer:
The spring constant \(k\) is approximately \(741.51 \, \text{N/m}\).

Would you like more details on any specific part of the calculation or have further questions?

### Relative Questions:
1. How would the spring constant change if the ball had a different mass?
2. How does the height from which the ball is dropped affect the compression of the spring?
3. What would happen if the spring had a non-negligible mass?
4. How would the answer differ if the gravitational acceleration was not \(9.8 \, \text{m/s}^2\)?
5. What other methods can be used to measure or calculate the spring constant in practice?

### Tip:
In energy conservation problems, always ensure that all forms of energy involved (potential, kinetic, elastic) are considered, and that no energy is lost to factors like friction or air resistance unless explicitly stated.

A 147 g ball is dropped from a height of 52.2 cm above a spring, which is compressed by 4.50803 cm. Given the acceleration of gravity as 9.8 m/s^2, calculate the spring constant k in N/m.

Learn how to calculate the spring constant of a spring compressed by a ball dropped from a height. This physics problem involves key concepts of energy conservation, gravitational potential energy, and elastic potential energy. Suitable for high school students in Grades 10-12.

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