A spring stores potential energy U0 when it is compressed a distance x0 from its uncompressed length. (a) In terms of U0, how much energy does the spring store when it is compressed (i) twice as much and (ii) half as much? (b) In terms of x0, how much must the spring be compressed from its uncompressed length to store (i) twice as much energy and (ii) half as much energy?

Solution 17E When the spring gets compressed by a distance of x , the potential energy it stores is U 0 0 , which is equal to, U = 1 K x --------------------(1) 0 2 0 This is the potential energy of a spring compressed by x . 0 a) i) when the spring gets compressed twice as much, U = 1 K x = 1K (2x ) = 1 K × 4 x 2 2 2 0 2 0 = 4U 0 The potential energy would be 4 times as before. ii) When the spring gets compressed to half as much, 1 2 1 2 1 2 U = 2 K x = 2K (0.5x 0 = 2K × 0.25 × x 0 1 = U4. 0 The potential energy stored will be U . 4 0 b) i) to get a potential energy of 2U , the compressing must be, 0 1 2 1 2 2U =0 2 K x = 2 × 2 K x0 = K x 02 If we take a suitable x which will be 2x , then, 0 1 2 1 2 1 2 2 2U =0 2 K x = 2 K (2 x 0 = 2K × 2 x 0 = K x 0 The spring must be compressed by 2times the original compression to get twice potential energy. ii) to get a potential energy of 1/2U , the compressing must be, 0 2 2 1/2U =0 2K x = 2 × 2K x 0 = K4x 02 1 If we take a suitable x which will be x0, then, 2 x 2 1/2 U = 1 K x =2 1 K ( x ) =2 1 K × ( 0) = K x 2 0 2 2 2 0 2 2 4 0 The spring must be compressed by 1 times the original compression to get half of the 2 potential energy as before.