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4.3Newton’s Second Law
What about cases where the total force on an object is not zero, so that
Newton’s first law doesn’t apply. The object will have an acceleration. The
way we’ve defined positive and negative signs of force and acceleration
guarantees that positive forces produce positive accelerations, and likewise
for negative values. How much acceleration will it have. It will clearly
depend on both the object’s mass and on the amount of force.
Experiments with any particular object show that its acceleration is
directly proportional to the total force applied to it. This may seem wrong,
since we know of many cases where small amounts of force fail to move an
object at all, and larger forces get it going. This apparent failure of propor-
tionality actually results from forgetting that there is a frictional force in
addition to the force we apply to move the object. The object’s acceleration
is exactly proportional to the total force on it, not to any individual force on
it. In the absence of friction, even a very tiny force can slowly change the
velocity of a very massive object.
Experiments also show that the acceleration is inversely proportional to
the object’s mass, and combining these two proportionalities gives the
following way of predicting the acceleration of any object:
Newton’s Second Law
a=F
total
/m ,
where
m is an object’s mass
F
total
is the sum of the forces acting on it, and
a is the acceleration of the object’s center of mass.
We are presently restricted to the case where the forces of interest are
parallel to the direction of motion.
Example: an accelerating bus
Question: A VW bus with a mass of 2000 kg accelerates from 0
to 25 m/s (freeway speed) in 34 s. Assuming the acceleration is
constant, what is the total force on the bus.
Solution: We solve Newton’s second law for F
total
=ma, and
substitute
.
v/
.
t for a, giving
F
total
= m
.
v/
.
t
= (2000 kg)(25 m/s - 0 m/s)/(34 s)
= 1.5 kN .
A generalization
As with the first law, the second law can be easily generalized to include
a much larger class of interesting situations:
Suppose an object is being acted on by two sets of forces, one set
lying along the object’s initial direction of motion and another set
acting along a perpendicular line. If the forces perpendicular to the
initial direction of motion cancel out, then the object accelerates
along its original line of motion according to a=F
total
/m.
The relationship between mass and weight
Mass is different from weight, but they’re related. An apple’s mass tells
Chapter 4Force and Motion