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A spring of force constant k is stretched a certain distance. It takes twice as much work to stretch a second spring by half this distance. What is the force constant of the second spring? (in units of k)
Correct answer is '8'. Can you explain this answer?
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A spring of force constant k is stretched a certain distance. It takes...
To solve this problem, let's consider the work done on each spring separately and then equate the two expressions to find the relationship between their force constants.
Let's assume that the first spring is stretched by a distance x. The work done on the spring can be calculated using the formula:
Work = (1/2)kx^2
where k is the force constant of the spring.
Now, let's consider the second spring. It is stretched by half the distance of the first spring, which is (1/2)x. The work done on this spring can be calculated using the same formula:
Work = (1/2)k'(1/2x)^2 = (1/2)k'(1/4)x^2 = (1/8)k'x^2
where k' is the force constant of the second spring.
According to the problem, it takes twice as much work to stretch the second spring compared to the first spring. Mathematically, this can be expressed as:
(1/8)k'x^2 = 2(1/2)kx^2
Simplifying this equation, we get:
(1/8)k' = k
Multiplying both sides by 8, we have:
k' = 8k
Therefore, the force constant of the second spring (k') is 8 times the force constant of the first spring (k). Hence, the correct answer is 8.
Let's assume that the first spring is stretched by a distance x. The work done on the spring can be calculated using the formula:
Work = (1/2)kx^2
where k is the force constant of the spring.
Now, let's consider the second spring. It is stretched by half the distance of the first spring, which is (1/2)x. The work done on this spring can be calculated using the same formula:
Work = (1/2)k'(1/2x)^2 = (1/2)k'(1/4)x^2 = (1/8)k'x^2
where k' is the force constant of the second spring.
According to the problem, it takes twice as much work to stretch the second spring compared to the first spring. Mathematically, this can be expressed as:
(1/8)k'x^2 = 2(1/2)kx^2
Simplifying this equation, we get:
(1/8)k' = k
Multiplying both sides by 8, we have:
k' = 8k
Therefore, the force constant of the second spring (k') is 8 times the force constant of the first spring (k). Hence, the correct answer is 8.
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A spring of force constant k is stretched a certain distance. It takes...
The correct answer is: 8
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A spring of force constant k is stretched a certain distance. It takes twice as much work to stretch a second spring by half this distance. What is the force constant of the second spring? (in units of k)Correct answer is '8'. Can you explain this answer?
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