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A string of length L and force constant ...

A string of length L and force constant k is stretched to obtain extension l. It is further stretched to obtain extension `l_(1)`. The work done in second stretching is

A

`(1)/(2)kl_(1)(2l+l_(1))`

B

`(1)/(2)kl_(1)^(2)`

C

`(1)/(2)k(l^(2)+l_(1)^(2))`

D

`(1)/(2)k(l_(1)^(2)-l^(2))`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem of finding the work done in the second stretching of a string, we can follow these steps: ### Step 1: Understand the Work Done in Stretching a String The work done (W) in stretching a string to an extension \( x \) is given by the formula: \[ W = \frac{1}{2} k x^2 \] where \( k \) is the force constant of the string. ### Step 2: Calculate the Work Done for the First Extension For the first extension \( l \), the work done \( W_1 \) is: \[ W_1 = \frac{1}{2} k l^2 \] ### Step 3: Define the Total Extension After Further Stretching After the first extension \( l \), the string is further stretched to a new extension \( l_1 \). The total extension after the second stretching becomes: \[ l' = l + l_1 \] ### Step 4: Calculate the Work Done for the Total Extension The work done \( W_2 \) to stretch the string to the total extension \( l' \) is: \[ W_2 = \frac{1}{2} k (l + l_1)^2 \] ### Step 5: Find the Work Done in the Second Stretching The work done in the second stretching (from extension \( l \) to \( l' \)) is the difference between the work done to stretch to \( l' \) and the work done to stretch to \( l \): \[ W_{\text{second}} = W_2 - W_1 \] Substituting the expressions for \( W_2 \) and \( W_1 \): \[ W_{\text{second}} = \frac{1}{2} k (l + l_1)^2 - \frac{1}{2} k l^2 \] ### Step 6: Simplify the Expression Now, we can simplify the expression: \[ W_{\text{second}} = \frac{1}{2} k \left[(l + l_1)^2 - l^2\right] \] Expanding \( (l + l_1)^2 \): \[ (l + l_1)^2 = l^2 + 2ll_1 + l_1^2 \] Thus, \[ W_{\text{second}} = \frac{1}{2} k \left[l^2 + 2ll_1 + l_1^2 - l^2\right] \] The \( l^2 \) terms cancel out: \[ W_{\text{second}} = \frac{1}{2} k (2ll_1 + l_1^2) \] ### Final Result Therefore, the work done in the second stretching is: \[ W_{\text{second}} = \frac{1}{2} k l_1 (2l + l_1) \]

To solve the problem of finding the work done in the second stretching of a string, we can follow these steps: ### Step 1: Understand the Work Done in Stretching a String The work done (W) in stretching a string to an extension \( x \) is given by the formula: \[ W = \frac{1}{2} k x^2 \] where \( k \) is the force constant of the string. ...
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Knowledge Check

  • A wire is stretched by one mm by a force of 1 kN. The work done is stretching the wire is

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