To understand why the stiffness of the second spring is the same as the first spring, we need to analyze the factors that contribute to the stiffness of a spring.

1. Stiffness of a spring:
The stiffness of a spring is a measure of its ability to resist deformation under an applied load. It is represented by the spring constant, denoted as K. The higher the value of K, the stiffer the spring.

2. Hooke's Law:
The stiffness of a spring is directly proportional to its material and geometric properties according to Hooke's Law. Mathematically, it can be expressed as:
F = K * x
where F is the applied force, K is the spring constant, and x is the displacement or deformation of the spring.

3. Geometric properties of a spring:
The geometric properties of a spring include wire diameter (d), mean coil diameter (D), and number of turns (N). These properties determine the physical dimensions and shape of the spring.

- Wire diameter (d): It is the diameter of the wire used to make the spring.
- Mean coil diameter (D): It is the average diameter of the coils in the spring.
- Number of turns (N): It is the total number of coils in the spring.

4. Relationship between geometric properties and stiffness:
The stiffness of a spring is influenced by its geometric properties as follows:

- Wire diameter (d): Increasing the wire diameter increases the stiffness of the spring. This is because a thicker wire can withstand higher loads without significant deformation.
- Mean coil diameter (D): Increasing the mean coil diameter decreases the stiffness of the spring. This is because a larger coil diameter allows for more displacement or deformation under the same applied load.
- Number of turns (N): Increasing the number of turns increases the stiffness of the spring. This is because more turns result in a longer spring length, which increases the overall stiffness.

5. Comparison of the first and second springs:
In the given scenario, the second spring is made of the same steel as the first spring, but with doubled values for wire diameter (2d), mean coil diameter (2D), and number of turns (2N).

- Wire diameter: The doubled value of wire diameter (2d) means the second spring has a thicker wire compared to the first spring.
- Mean coil diameter: The doubled value of mean coil diameter (2D) means the second spring has larger coils compared to the first spring.
- Number of turns: The doubled value of the number of turns (2N) means the second spring has twice as many coils compared to the first spring.

6. Impact of changes on stiffness:
Considering the impact of these changes on the stiffness of the second spring:

- Wire diameter: The increased wire diameter (2d) of the second spring increases its stiffness compared to the first spring.
- Mean coil diameter: The increased mean coil diameter (2D) of the second spring decreases its stiffness compared to the first spring.
- Number of turns: The increased number of turns (2N) of the second spring increases its stiffness compared to the first spring.

7. Conclusion:
The increase in stiffness due to the increased wire diameter and number of turns is offset by the decrease in stiffness due to the increased mean coil diameter. Therefore, the stiffness of the second spring remains the same as the first spring,