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0593_C10_fm Page 338 Monday, May 6, 2002 2:57 PM

338 Dynamics of Mechanical Systems

k

B
m
FIGURE 10.9.2
Y
A vertical mass–spring system.
and the block, with the sum of the potential energy of the spring and the kinetic energy
of the block being constant. (We will discuss potential energy in the next chapter.)
As another example of work–energy transfer, consider the mass–spring system arranged
vertically as in Figure 10.9.2. Suppose B is held in a position where the spring is
unstretched. If B is then released from rest from this position, it will fall and stretch the
spring and eventually come to rest at an extreme downward position. Questions arising
then include how far does B fall and what is the spring force when B reaches this maximum
downward displacement? To answer these questions, consider that as B falls, the weight
(or gravity) force on B is in the direction of the movement of B, whereas the spring force
on B is opposite to the movement of B. Because B is at rest at both the beginning and the
end of the movement, there is no change in the kinetic energy of B. The net work on B is
then zero. That is,

W == mgd −( ) 2 kd 2 (10.9.8)
0
1
where d is the distance B moves downward. By solving for d we obtain:

d = 2 mg k (10.9.9)

The spring force in this extended position is, then,

F = kd = 2 mg (10.9.10)

The result of Eq. (10.9.10) shows that a suddenly applied weight load on a spring creates
a force twice that of the weight. This means that if a weight is suddenly placed on a
machine or structure the force generated is twice that required to support the weight in
a static equilibrium configuration.

10.10 Skidding Vehicle Speeds: Accident Reconstruction Analysis

The work–energy principle is especially useful in determining speeds of accident vehi-
cles by using measurements of skid-mark data. Indeed, the work–energy principle
together with the conservation of momentum principles are the primary methods used
by accident reconstructionists when attempting to determine vehicle speeds at various
stages of an accident.