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The potential energy of a long spring when stretched by 2cm is U. If the spring is stretched by 8 cm, the potential energy stored in it is
- a)8 U
- b)16 U [2006]
- c)U/ 4
- d)4 U
Correct answer is option 'B'. Can you explain this answer?
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To understand why the correct answer is option 'B', let's analyze the concept of potential energy stored in a spring.
1. Understanding Potential Energy in a Spring:
When a spring is stretched or compressed, it stores potential energy due to the deformation. This potential energy can be calculated using Hooke's Law and the formula for potential energy.
2. Hooke's Law:
Hooke's Law states that the force required to stretch or compress a spring is directly proportional to the displacement from its equilibrium position. Mathematically, it can be represented as:
F = -kx
Where F is the force applied, k is the spring constant, and x is the displacement from the equilibrium position.
3. Potential Energy Formula:
The potential energy stored in a spring can be calculated using the formula:
U = (1/2) kx^2
Where U is the potential energy, k is the spring constant, and x is the displacement from the equilibrium position.
4. Applying the Given Information:
According to the question, the potential energy of the spring when stretched by 2 cm is U. Let's denote this as U1. We need to find the potential energy when the spring is stretched by 8 cm, denoted as U2.
Given: x1 = 2 cm, U1 = U
To find: U2
We know that potential energy is directly proportional to the square of the displacement. Therefore, we can write:
U1/U2 = (x1^2)/(x2^2)
Substituting the given values:
U/U2 = (2^2)/(8^2)
U/U2 = 1/16
Cross-multiplying:
U2 = 16U
Hence, the potential energy stored in the spring when stretched by 8 cm is 16U. Therefore, the correct answer is option 'B'.
1. Understanding Potential Energy in a Spring:
When a spring is stretched or compressed, it stores potential energy due to the deformation. This potential energy can be calculated using Hooke's Law and the formula for potential energy.
2. Hooke's Law:
Hooke's Law states that the force required to stretch or compress a spring is directly proportional to the displacement from its equilibrium position. Mathematically, it can be represented as:
F = -kx
Where F is the force applied, k is the spring constant, and x is the displacement from the equilibrium position.
3. Potential Energy Formula:
The potential energy stored in a spring can be calculated using the formula:
U = (1/2) kx^2
Where U is the potential energy, k is the spring constant, and x is the displacement from the equilibrium position.
4. Applying the Given Information:
According to the question, the potential energy of the spring when stretched by 2 cm is U. Let's denote this as U1. We need to find the potential energy when the spring is stretched by 8 cm, denoted as U2.
Given: x1 = 2 cm, U1 = U
To find: U2
We know that potential energy is directly proportional to the square of the displacement. Therefore, we can write:
U1/U2 = (x1^2)/(x2^2)
Substituting the given values:
U/U2 = (2^2)/(8^2)
U/U2 = 1/16
Cross-multiplying:
U2 = 16U
Hence, the potential energy stored in the spring when stretched by 8 cm is 16U. Therefore, the correct answer is option 'B'.
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The potential energy of a long spring when stretched by 2cm is U. If the spring is stretched by 8 cm, the potential energy stored in it isa)8 Ub)16 U [2006]c)U/ 4d)4 UCorrect answer is option 'B'. Can you explain this answer?
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