Question

If the work done in strectching a wire by 1mm is 2J, then work necessary for stretching another wire of same material but with double radius of cross -section and half of the length by 1 mm is

Answer 1

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Answer 2

Thus the work done is 16 J

Explanation:

The work done in stretching a wire is given as

W = (1/2).F.dl

F is the force required

dl is the increase in length (which remains same in both cases)

So, we can have

W1 / W2 = [(1/2).F1.dl] / [(1/2).F2.dl]

2J / W2 = F1 / F2

W2 =2 x ( F2/F1)  (1)

Now we know that the force is given as

F = Y.A.(dl/l)

here  

Y is the Young's Modulus

A is the area of cross-section of the wire  and l is the length of the wire

F = Y.(πr2).(dl/l)

Now, in this case we will have

F1/F2 =  [Y.(πr12).(dl/l1)] / [Y.(πr22).(dl/l2)]

F1/F2 = (r12 / r22) . (l2/l1)  (2)

Now as per given

r2 = 2r1

l2= l1/2 (not energy)

We have

F1/F2 = (1/4).(1/2)

or

F1/F2 = 1/8

So, by using (1) we get

W2 = 2 x ( F2/F1)  = 2 J x 8

Thus, work done will be

W2 = 16 J