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A spring with spring constant k when stretched through 1 cm has a potential energy U. If it is stretched by 4 cm, the potential energy will become
- a)4U
- b)16U
- c)8U
- d)32U
Correct answer is option 'B'. Can you explain this answer?
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Explanation:
To understand why the correct answer is option 'B', let's look at the concept of potential energy stored in a spring.
1. Potential Energy in a Spring:
When a spring is stretched or compressed, it stores potential energy. This potential energy is given by the formula:
U = (1/2) * k * x^2
Where U is the potential energy, k is the spring constant, and x is the displacement of the spring from its equilibrium position.
2. Given Information:
In this question, it is given that when the spring is stretched through 1 cm, it has a potential energy U. Let's denote this displacement as x1 = 1 cm.
We need to find the potential energy when the spring is stretched by 4 cm, which can be denoted as x2 = 4 cm.
3. Calculation:
Using the formula for potential energy in a spring, we can compare the potential energies at the two different displacements.
U1 = (1/2) * k * x1^2
U2 = (1/2) * k * x2^2
Substituting the given values:
U1 = (1/2) * k * (1 cm)^2
U2 = (1/2) * k * (4 cm)^2
Simplifying further:
U1 = (1/2) * k * 1 cm^2
U2 = (1/2) * k * 16 cm^2
U2 = 8 * (1/2) * k * 1 cm^2
U2 = 8 * U
Therefore, the potential energy when the spring is stretched by 4 cm is 8 times the potential energy when it is stretched by 1 cm. In other words, the potential energy becomes 8U.
Hence, the correct answer is option 'B' - 16U.
To understand why the correct answer is option 'B', let's look at the concept of potential energy stored in a spring.
1. Potential Energy in a Spring:
When a spring is stretched or compressed, it stores potential energy. This potential energy is given by the formula:
U = (1/2) * k * x^2
Where U is the potential energy, k is the spring constant, and x is the displacement of the spring from its equilibrium position.
2. Given Information:
In this question, it is given that when the spring is stretched through 1 cm, it has a potential energy U. Let's denote this displacement as x1 = 1 cm.
We need to find the potential energy when the spring is stretched by 4 cm, which can be denoted as x2 = 4 cm.
3. Calculation:
Using the formula for potential energy in a spring, we can compare the potential energies at the two different displacements.
U1 = (1/2) * k * x1^2
U2 = (1/2) * k * x2^2
Substituting the given values:
U1 = (1/2) * k * (1 cm)^2
U2 = (1/2) * k * (4 cm)^2
Simplifying further:
U1 = (1/2) * k * 1 cm^2
U2 = (1/2) * k * 16 cm^2
U2 = 8 * (1/2) * k * 1 cm^2
U2 = 8 * U
Therefore, the potential energy when the spring is stretched by 4 cm is 8 times the potential energy when it is stretched by 1 cm. In other words, the potential energy becomes 8U.
Hence, the correct answer is option 'B' - 16U.
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