Spring Potential Energy

POTENTIAL ENERGY FUNCTIONS

Potential energy is clearly a function of position. We now find this function for the (approximately constant) gravitational force near the surface of the earth and for the force exerted by an ideal spring.

Gravitational Potential Energy (Near the Earth's Surface)

From Eq. 4.2, the work done by gravity on a particle of mass m whose vertical coordinate changes from y_f to y _i is

W _g = −mg (y _f - y _i )

We have W _g = −∆ U_g =− (U_f − U _i ). We deduce that the gravita­tional potential energy near the surface of the earth is given by

U g = mgy (4.17)

Implicit in this expression is the choice U_g = 0 at у = 0.

FIGURE 4.7

The work done by the force due to a spring when the displacement of the free end changes from x _i to x _f is given by Eq. 4.3:

From W_sp = −∆U_sp = − (U_f − U _i ), we deduce that the potential energy of the spring is

Note that U _sp = 0 at x = 0. Since Hooke's law is also valid for compressions, the same potential energy function, shown in Fig. 4.7, applies for negative x (so long as the coils do not touch).

CHAPTER 5

Conservations Lows in Mechanics

Дата добавления: 2014-01-05