Page 358 - Dynamics of Mechanical Systems

0593_C10_fm Page 339 Monday, May 6, 2002 2:57 PM

Introduction to Energy Methods 339

To illustrate the procedure, suppose an automobile leaves skid marks from all four
wheels in coming to an emergency stop. Given the length d of the skid marks, the objective
is to determine the vehicle speed when the marks first began.
Skid marks are created by abrading and degrading tires sliding on a roadway surface.
The tire degradation is due to friction forces and heat abrading the rubber. The friction
forces are proportional to the normal (perpendicular to the roadway surface) forces on
the tires and to the coefficient of friction µ. The friction coefficient, ranging in value from
0 to 1.0, is a measure of the relative slipperiness between the tires and the roadway
pavement. If F and N are equivalent friction and normal forces (see Section 6.5), they are
related by the expression:

F =µ N (10.10.1)

If an automobile is sliding on a flat, level (horizontal) roadway, a free-body diagram of
the vehicle shows that the normal force N is equal to the vehicle weight w. Then, as the
vehicle slides a distance d, the work W done by the friction force (acting opposite to the
direction of the sliding) is:

W =− Fd =−µ Nd =−µ wd =−µ mgd (10.10.2)

where m is the mass of the automobile.
Let v be the desired speed of the automobile when the skid marks first appear. Then,
the kinetic energy K of the vehicle at that point is:
i
K = ( ) 2
i 12 mv (10.10.3)

Because the kinetic energy K at the end of the skid marks is zero (the vehicle is then
f
stopped), the work energy principle produces:

W = ∆ K = K − K or − µ mgd = −( ) 2 mv 2 (10.10.4)
0
1
f i
or

v = 2µ gd (10.10.5)

Observe that the calculated speed is independent of the automobile mass.
To illustrate how the work–energy principle may be used in conjunction with the
momentum conservation principles, suppose an automobile with mass m slides a distance
1
d before colliding with a stopped automobile having mass m . Suppose further that the
2
1
two vehicles then slide together (a plastic collision) for a distance d before coming to rest.
2
The questions arising then are what were the speeds of the vehicles just before and just
after impact and what was the speed of the first vehicle when it first began to slide?
To answer these questions, consider first from Eq. (10.10.5) that the speed v of the
a
vehicles just after impact is:
v = 2µ gd (10.10.6)
a 2