When a long spring is stretched, it stores potential energy due to the work done in stretching it. The amount of potential energy stored in the spring depends on the amount of stretch or displacement of the spring from its equilibrium position.

Given:
- The potential energy of the spring when stretched by 2 cm is U.

To find:
- The potential energy stored in the spring when stretched by 10 cm.

Let's analyze the situation step by step:

Understanding the potential energy of a stretched spring:

When a spring is stretched, it follows Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement from the equilibrium position. Mathematically, this relationship can be expressed as:

F = -kx

where F is the force exerted by the spring, k is the spring constant, and x is the displacement of the spring.

The potential energy stored in a spring can be calculated using the formula:

U = (1/2)kx^2

where U is the potential energy and x is the displacement.

Comparing the two cases:

Case 1: When the spring is stretched by 2 cm, the potential energy is U.

Case 2: When the spring is stretched by 10 cm, we need to find the potential energy.

Let's denote the potential energy in case 2 as U'.

Using the formula for potential energy, we can write:

U' = (1/2)kx'^2

where x' is the displacement of the spring in case 2.

Using proportionality:

From Hooke's Law, we know that the force exerted by the spring is directly proportional to the displacement. Therefore, we can write:

F = kx

In case 1, the force exerted by the spring when stretched by 2 cm is:

F = k(2 cm) = 2k

In case 2, the force exerted by the spring when stretched by 10 cm is:

F' = k(10 cm) = 10k

Using the work-energy principle:

The work done in stretching the spring is equal to the potential energy stored in it. According to the work-energy principle, the work done is equal to the force applied multiplied by the displacement. Therefore, we can write:

Work = Force * Displacement

In case 1, the work done is:

Work = (2k) * (2 cm) = 4k cm

In case 2, the work done is:

Work' = (10k) * (10 cm) = 100k cm

Comparing the potential energies:

Since the potential energy is equal to the work done, we can write:

U = Work = 4k cm

U' = Work' = 100k cm

To find the ratio of U' to U, we can divide the two expressions:

U'/U = (100k cm) / (4k cm) = 25

Therefore, the potential energy stored in the spring when stretched by 10 cm is 25 times the potential energy when stretched by 2 cm.

Conclusion:

The correct answer is option D) 25U.