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Explain the elastic potential energy of spring and obtain an expression for this energy . (a) |
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Answer» Solution :Consider an elastic spring ,obeying Hook.s law with NEGLIGIBLE mass whose one end is tied rigidly to a wall as shown in figure . At the other end of the spring a block has been tied and it restring on a smooth horizontal surface . We shall for the sake of SIMPLICITY restrict the motion of the block in the X - direction . In the normal position of the spring , the position of the block is taken as x=0 ,this is shown as figure (a) When the block is pulled and the length of spring + x is INCREASED a restoring force `F_(S)` is producedin the springwhich tries to bring the spring back to its normal position .the restoring force (spring force ) is produced when the spring is COMPRESSED .This is shown in figure (b) and (c) . The restoring force is directly proportional to the change in the length of the spring and is in the direction opposite to the change in the length . This law of force for the spring is called Hook.s law . ` :. F_(S) = - kx ` where k is the springconstant or force constant of the spring Spring constant`k = (F_(S))/x` (magnitude ) , its unit is `Nm^(-1)` The spring is said to be stiff if k is large and soft if it is small . Suppose that the block is pull outwards as in figure (b) . if the extension is `x_(m)` the work done by the spring force is , `W_(S) =int_(0)^(x_(m)) F_(S)dx = - int_(0)^(x_(m)) ""[ :. F_(S)=- kDeltax]` `=-k[(x^(2))/2]_(0)^(x_(m))` `=k[(x^(2))/2-0]` `=-(kx_(m)^(2))/2`  This expression may also be obtained by considereing the area of the triangle as in figure (d). The work done by the external PULLING force F is positive since it overcomes the spring force . If spring is compressed as shown in figure (c) with a displacement `x_(c) ( lt 0)` .The spring force dows work `W_(S) = - (kc_(c)^(2))/2` while the external force F does work + `(kx_(c)^(2))/2` `W_(S) = - int_(x_(i))^(x_(f))" kx dx" = - k [(x^(2))/2]_(x_(i))^(x_(f))` `W_(S) =(kx_(i)^(2))/2 - (kx_(f)^(2))/2 ` Hence , the work done by the spring force depends only on the end points . If the block is pulled from `x_(i)` and allowed to return to`x_(i)` the work done by spring force , `W_(S) =-int_(x_(i))^(x_(i))" kx dx" = - k int_(x_(i))^(x_(i)) "x dx" = - k [(x^(2))/2]_(x_(i))^(x_(i))` ` = - k[ (x_(i)^(2))/2-(x_(i)^(2))/2]` `= 0 ` Hence , the work done by the spring force in a cycle process is zero and so spring force is conservative . |
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