120 PHYSICS • not available explicitly. Another observation is that Newton’s second law for two or three dimensions is in vector form whereas the work-energy theorem is in scalar form. In the scalar form, information with respect to directions contained in Newton’s second law is not present. Example 6.6 A block of mass m = 1 kg, moving on a horizontal surface with speed v = 2 ms–1 enters a rough patch ranging i from x = 0.10 m tox = 2.01 m. The retarding forceF on the block in this range is inverselyrproportional to x over this range, −k Fr = for 0.1 < x < 2.01 m x = 0 for x < 0.1m and x > 2.01 m where k = 0.5 J. What is the final kinetic energy and speed vf of the block as it crosses this patch ? Answer From Eq. (6.8a) 2.01 k KK dxfi x0.1 1 2 2.01 mv k ln x 0.1 2 i 1 2 mv k ln 2.01/0.1 2 i= 2 − 0.5 ln (20.1) = 2 − 1.5 = 0.5 J vf = 2Kf /m = 1ms −1 Here, note that ln is a symbol for the natural logarithm to the base e and not the logarithm to the base 10 [ln X = loge X = 2.303 log10 X]. • 6.7 THE CONCEPT OF POTENTIAL ENERGY The word potential suggests possibility or capacity for action. The term potential energy brings to one’s mind ‘stored’ energy. A stretched bow-string possesses potential energy. When it is released, the arrow flies off at a great speed. The earth’s crust is not uniform, but has discontinuities and dislocations that are called fault lines. These fault lines in the earth’s crust are like ‘compressed springs’. They possess a large amount of potential energy. An earthquake results when these fault lines readjust. Thus, potential energy is the ‘stored energy’ by virtue of the position or configuration of a body. The body left to itself releases this stored energy in the form of kinetic energy. Let us make our notion of potential energy more concrete. The gravitational force on a ball of mass m is mg . g may be treated as a constant near the earth surface. By ‘near’ we imply that the height h of the ball above the earth’s surface is very small compared to the earth’s radius RE (h <

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