1 Potential energy This is the energy possessed by a body by virtue of its position. It is a scalar function associated with a conservative force. The potentials energy U(x,y,z) depends only on the position of an object. Its SI unit is joule. If the potential energy at y = 0 is 0, then the potential energy at any height y is U(y) = mgy. For a conservative one-dimensional force F(x), the work done in moving from x1 to x2 is x2 W(x1,x2) = This requires that From the work-energy theorem, we then have that x1 F(x)dx = U(x1)−U(x2) = −[U(x2)−U(x1)]. F(x) = −dU dx W(x1,x2) = 1 2mv22 − 1 U(x1) + 1 2mv21, 2mv21 = U(x2)+ 1 2mv22. This is the law of conservation of energy. It shows that U + 1 (52) (53) (54) (55) (56) 2mv2 = E stays constant. This quantity is called the total mechanical energy of the object E =U+1 2mv2 =constant. (57) Example: A skier passes over the crest of a small hill at a speed of 3.6 m/s. How fast will she be moving when she has dropped to a point 5.6 m lower than the crest of the hill? Neglect friction and take g to be 9.8 m/s2. Solution: P.E1 +K.E1 = P.E2 +K.E2 mgh1 + 1 2mv21 = mgh2 + 1 2mv22 v2 = v21 + 2g(h1 −h2) = 3.62 + 2(9.8)(5.6) = 11.1m/s.
