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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...
The correct answer is: 8
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A spring of force constant k is stretched a certain distance. It takes...
Given Information:
- Force constant of the first spring: k
- Work done to stretch the first spring: W
- Distance stretched by the first spring: d
Calculating the Force Constant for the Second Spring:
Let's assume the force constant of the second spring is K.
According to Hooke's Law, the force required to stretch a spring is given by:
F = kx
where F is the force applied, k is the force constant, and x is the displacement from the equilibrium position.
The work done to stretch a spring is given by the formula:
W = (1/2)kx^2
For the First Spring:
The work done to stretch the first spring is given as W.
Therefore, the equation becomes:
W = (1/2)kd^2
For the Second Spring:
The distance stretched by the second spring is half of the distance stretched by the first spring, which is d/2.
The work done to stretch the second spring is twice that of the first spring, which is 2W.
Therefore, the equation becomes:
2W = (1/2)K(d/2)^2
Simplifying this equation:
2W = (1/8)Kd^2
Multiplying both sides by 8:
16W = Kd^2
Comparing the Equations:
We can now compare the equations for the first spring and the second spring:
W = (1/2)kd^2
16W = Kd^2
Since the work done is the same for both springs (W = 16W), we can equate the equations:
(1/2)kd^2 = Kd^2
Dividing both sides by d^2:
(1/2)k = K
Therefore, the force constant of the second spring, K, is 8 times the force constant of the first spring, k.
Conclusion:
The force constant of the second spring is 8 times the force constant of the first spring.
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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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