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A spring of constant 5×103Nm1 is stretched initially by 5cm from the unstretched position. Find the work required to stretch it further by another 5cm.
A) 6.25Nm
B) 12.5Nm
C) 18.75Nm
D) 25.00Nm

Answer
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Hint: Work done in stretching a spring is stored as the elastic potential energy given by the equation,
U=12kx2
Where k is the spring constant and x is the displacement.
The change in potential energy of an object between two positions is equal to the work done in moving the object from first position to next. So, in order to calculate the work done to move spring from one position to another, it is enough to find the difference in potential energy between two positions. This work depends upon the spring constant and the distance stretched.
Therefore,
W=UfUi
Ui is the initial energy given as,
Ui=12kxi2
Where xi is the initial displacement.
Uf is the final energy given as,
Uf=12kxf2
Where xf is the final displacement.

Complete step by step solution:
Work done in stretching a spring is stored as the elastic potential energy given by the equation,
U=12kx2
Where k is the spring constant and x is the displacement.
This potential energy can be said as the energy due to the deformation of the spring
Given,
Initial displacement,
xi=5cm5×102m0.05m
Final displacement,
xf=0.05+0.050.1m
Value of spring constant, k=5×103Nm1
The change in potential energy of an object between two positions is equal to the work done in moving the object from first position to next. So, in order to calculate the work done to move spring from one position to another, it is enough to find the difference in potential energy between two positions. This work depends upon the spring constant and the distance stretched.
Therefore, the work required to stretch it further by another 5cm is the change in initial potential energy and final potential energy.
W=UfUi ………… (1)
Where Ui is the initial energy given as,
Ui=12kxi2
Substituting the given values, we get
Ui=12×5×103×(0.05)2=6.25Nm
Uf is the final energy given as,
Uf=12kxf2
Substituting the given values, we get
Uf=12×5×103×(0.1)2=25Nm
Now substitute the initial and final energies in equation (1).
W=UfUi25Nm6.5Nm18.5Nm
Therefore, the work required to stretch it further by another 5cm is 18.5Nm.

So, the answer is option C.

Note: Here, while substituting for the value of displacement in the final potential energy remember to substitute the value 0.1m. The spring was already stretched by 0.05m and we need to find the work done when it is further stretched by 0.05m.So the final displacement will be 0.05m+0.05m=0.1m