A Classical Approach of Newtonian Mechanics

5 CONSERVATION OF ENERGY 5.3 Work

∫

m

dt^2

dx = m

dt^2 dt

dt = (^) dt
2
(^) dt, (5.20)
xA xB x ->
Figure 39: Work performed by a 1 - dimensional force
to xB? Well, a straight-forward application of Eq. (5.17) [with f = (f, 0, 0) and
dr = (dx, 0, 0)] yields
xB
W = f(x) dx. (5.18)
xA
In other words, the net work done by the force as the object moves from displace-
ment xA to xB is simply the area under the f(x) curve between these two points,
as illustrated in Fig. 39.
Let us, finally, round-off this discussion by re-deriving the so-called work-
energy theorem, Eq. (5.14), in 1-dimension, allowing for a non-constant force.
According to Newton’s second law of motion,
d^2 x
f = m
dt^2

. (5.19)

Combining Eqs. (5.18) and (5.19), we obtain

∫xB

d^2 x (^)
∫tB
d^2 x dx (^)
∫tB
d m
dx
! 2
where x(tA) = xA and x(tB) = xB. It follows that
W =
1
m v 2 −
1
m^ v^2 =^ ∆K,^ (5.21)^
2 B^2 A^
where vA = (dx/dt)t and vB = (dx/dt)t. Thus, the net work performed on a
body by a non-uniform force, as it moves from point A to point B, is equal to the
xA tA^ tA dt^
W =
f^ -

^
A B