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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