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A spring of force constant k is cut into two pieces such that one piece is double the length of the other. The force constant of the longer piece will be
- a)1.5k
- b)3k
- c)2k
- d)2/3 k
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
A spring of force constant k is cut into two pieces such that one pie...
Explanation:
When a spring is cut into two pieces, the force constant of the longer piece will be different from the force constant of the shorter piece. Let's denote the force constant of the original spring as k, and the lengths of the two pieces as L1 and L2, where L2 is double the length of L1.
Using Hooke's Law:
Hooke's Law states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. Mathematically, it can be expressed as:
F = -kx
Where F is the force applied, k is the force constant, and x is the displacement.
Relation between force constants and lengths:
The force constant of a spring is inversely proportional to its length. Mathematically, it can be expressed as:
k ∝ 1/L
Where k is the force constant and L is the length of the spring.
Relation between lengths and force constants:
If we consider the original spring with force constant k and length L, and then cut it into two pieces such that one piece is double the length of the other, we can establish the following relations:
L2 = 2L1
k2 ∝ 1/L2 (Force constant of longer piece)
k1 ∝ 1/L1 (Force constant of shorter piece)
Using the relation between lengths:
From the given information, we know that L2 = 2L1. Substituting this into the relation for force constants, we get:
k2 ∝ 1/(2L1)
Comparing force constants:
We can compare the force constants of the longer and shorter pieces by taking their ratio:
k2/k1 = (1/(2L1))/(1/L1) = (1/(2L1))*(L1/1) = 1/2
Therefore, the force constant of the longer piece is half of the force constant of the shorter piece.
Conclusion:
From the above comparison, we can see that the force constant of the longer piece is half of the force constant of the shorter piece. Therefore, the correct answer is option A: 1.5k.
When a spring is cut into two pieces, the force constant of the longer piece will be different from the force constant of the shorter piece. Let's denote the force constant of the original spring as k, and the lengths of the two pieces as L1 and L2, where L2 is double the length of L1.
Using Hooke's Law:
Hooke's Law states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. Mathematically, it can be expressed as:
F = -kx
Where F is the force applied, k is the force constant, and x is the displacement.
Relation between force constants and lengths:
The force constant of a spring is inversely proportional to its length. Mathematically, it can be expressed as:
k ∝ 1/L
Where k is the force constant and L is the length of the spring.
Relation between lengths and force constants:
If we consider the original spring with force constant k and length L, and then cut it into two pieces such that one piece is double the length of the other, we can establish the following relations:
L2 = 2L1
k2 ∝ 1/L2 (Force constant of longer piece)
k1 ∝ 1/L1 (Force constant of shorter piece)
Using the relation between lengths:
From the given information, we know that L2 = 2L1. Substituting this into the relation for force constants, we get:
k2 ∝ 1/(2L1)
Comparing force constants:
We can compare the force constants of the longer and shorter pieces by taking their ratio:
k2/k1 = (1/(2L1))/(1/L1) = (1/(2L1))*(L1/1) = 1/2
Therefore, the force constant of the longer piece is half of the force constant of the shorter piece.
Conclusion:
From the above comparison, we can see that the force constant of the longer piece is half of the force constant of the shorter piece. Therefore, the correct answer is option A: 1.5k.
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Community Answer
A spring of force constant k is cut into two pieces such that one pie...
The force constant of a spring is inversely proportional to its length. A spring of length L is cut into two pieces of lengths x and (L - x), such that x = 2 (L - x) or x = 2L/3
The force constant of the spring of length x is related to the force constant k of the complete spring of length L as,
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