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WORK, ENERGY AND POWER 81 external force F does work + kx / 2 . If the block and vice versa, however, the total mechanical c energy remains constant. This is graphically is moved from an initial displacement x toia depicted in Fig. 5.8. final displacement x ,fthe work done by the spring force W is s f k xi k xf W s − ∫ k x dx = − (5.17) xi 2 2 Thus the work done by the spring force depends only on the end points. Specifically, if the block is pulled from x and allowed to return to x ; i i xi 2 2 k i k xi W s= −∫k x d = 2 − 2 xi = 0 (5.18) Fig. 5.8 Parabolic plots of the potential energy V and The work done by the spring force in a cyclic kinetic energy K of a block attached to a spring obeying Hooke’s law. The two plots process is zero. We have explicitly demonstrated that the spring force (i) is position dependent are complementary, one decreasing as the other increases. The total mechanical only as first stated by Hooke, (F =s− kx); (ii) energy E = K + V remains constant. does work which only depends on the initial and final positions, e.g. Eq. (5.17). Thus, the spring ⊳Example5.8 Tosimulatecaraccidents,auto force is a conservative force. Wedefinethepotentialenergy V(x)ofthespring manufacturersstudythecollisionsofmoving carswithmountedspringsofdifferentspring to be zero when block and spring system is in the constants. Consideratypicalsimulationwith equilibrium position. For an extension (or compression) x the above analysis suggests that a car of mass 1000 kg moving with a speed 18.0 km/h on a smooth road and colliding 2 kx withahorizontallymountedspringofspring V(x) = 2 (5.19) constant 5.25 × 10 N m . What is the maximumcompressionofthespring ? You may easily verify that − dV/dx = − k x, the spring force. If the block of mass m in Fig. 5.7 is extended to x mnd released from rest, then its Answer At maximum compression the kinetic total mechanical energy at any arbitrary point x, energy of the car is converted entirely into the wherexliesbetween –x and +x willbegivenby m m, potential energy of the spring. The kinetic energy of the moving car is 1k x = k x + m v 2 2 m 2 2 1 2 K = 2 mv where we have invoked the conservation of mechanical energy. This suggests that the speed = 1 ×10 ×5×5 and the kinetic energy will be maximum at the 2 equilibrium position, x = 0, i.e., K = 1.25 × 10 J4 1 2 1 2 –1 –1 2 m vm = 2 k xm wherewehaveconverted18kmh to5ms [Itis useful to remember that 36 km h = 10 m s ]. –1 where v is the maximum speed. At maximum compression x , the potential m m energy V of the spring is equal to the kinetic v = k x energy K of the moving car from the principle of or m m m conservation of mechanical energy. Note that k/m has the dimensions of [T ] and our equation is dimensionally correct. The 1 2 V = k xm kinetic energy gets converted to potential energy 2 Reprint 2026-27