Force
Now we can say exactly what we mean by
the total force on a moving (or non moving)
object. By definition
The total force on an object =
(Its mass)x(Its acceleration)
This is usually called Newton’s second law
and is written F=ma. However it really is just
a definition of the total force on an object until we
say something later about the origin of forces.
A baseball player slides into third base.
What are the directions of his velocity,
acceleration and the total force on his body ?
velocity
acceleration
total force
a
toward 3rd toward 3rd
away from 3rd
b
toward 3rd
away from 3rd toward 3rd
c
toward 3rd
away from 3rd away from 3rd
d
away from 3rd toward 3rd
toward 3rd
e
toward 3rd
toward 3rd
toward 3rd
We now know what the total force on
an object is and we could calculate it if we knew
its mass and the position of the object at each
time in its motion (by calculating the acceleration
from the positions and the time intervals).
However this information would not let us
(or a professional engineer or scientist)
PREDICT what would happen to this object in
the future. For that we need a theory, sometimes
called a model, of what the force is.
Physicists, analysing experiments for over
3 centuries, have found that essentially
all the forces encountered in nature can
be modeled as
Gravitational
Electromagnetic or
Nuclear Forces
Our society uses all of these, but for most
of the course, and in most of everyday life,
we mainly encounter the first two.
Gravitational Force:
This is the force which makes objects fall toward
the earth when you drop them. Even when
studied at a very elementary level (as here)
the gravitational force has properties which make
it act quite differently from forces of the
electromagnetic or nuclear type.
To understand the essential feature, consider
the famous experiment done by Galileo more
than three hundred years ago:
Galilei Galileo lived in Italy from 1564 to 1642
His work on motion preceded Newton’s
theories and provided part of the basis for
them. He lived in Pisa, Italy where, among many
other scientific experiments, he studied the
time for dropped objects made of different masses
and materials to fall to earth. Some of these
experiments were performed by dropping objects
off the leaning tower of Pisa, a famous example
of bad engineering which is still standing (and
was not designed by Galileo).
An essential experimental finding of Galileo’s
experiments is that if only gravity acts on them,
objects of all masses drop toward the surface
of the earth at the same rate, so that if you drop
them from the same height at the same time,
they hit the ground at the same instant.
This had not been understood before and the reason
Galileo got it right (after hundreds of years of
philosophical speculation about it) is that he did
very careful experiments. In fact his first ideas
about how the objects would fall were wrong and
he had to revise them to make them consistent
with his experimental data.
What does this result of Galileo’s
tell us about the gravitational force?
Remember that F=ma or equivalently, a=F/m
So you might think that if the mass were twice
as big the acceleration would be half as
big. Which of the following resolves this
contradiction with Galileo’s experiments in
a logical way?
A. Newton’s 2nd law does not apply.
B. The acceleration is different but the
time for the drop is not.
C. The gravitational force on an object
doubles if its mass doubles.
D. The gravitational force on an object is
half as big if its mass doubles.
The conclusion is that the gravitational
Force doubles if the mass doubles so that the
mass cancels out in F=ma.
Fgravitational=m x something
where ‘something’ does not depend on
the mass. We put the notation ‘gravitational’
because this equation IS NOT TRUE FOR
FORCES WHICH ARE NOT GRAVITATIONAL
What is the ‘something’??
Near the surface of the earth (only)
we can find out by analysing data on a falling
object in more detail.
Here is some data, taken from such a falling
object
Speed
Distance fallen
Time
(m/sec)
(meters)
(sec)
0
0
0
0.033
0.017
0.515152
0.067
0
-0.51515
0.1
0.034
1.030303
0.133
0.06
0.787879
0.167
0.102
1.272727
0.2
0.144
1.272727
0.233
0.203
1.787879
0.267
0.254
1.545455
0.3
0.355
3.060606
0.333
0.457
3.090909
0.367
0.508
1.545455
Speed
(m/s)
4
3
2
data
1
straight line
0
-1
1
3
5
7
9
11
Time (1/30 sec)
Precise experiments always give a straight line
with the same slope corresponding to an
acceleration of about 9.8m/s2 (number depends
on altitude).
If you use data like this to calculate the
velocity you find out that the CHANGE in velocity
is the same in each instant of time. In terms
of Fgravitational=mx’something’
this experiment then tells us that
A.‘something’ is zero
B.‘something’ is independent of mass
C.‘something’ is does not depend on time
D.‘something’ increases with time.
C. The constant slope means that the
acceleration is not changing in time
B. Is true but this experiment gives us
no information about it.
We conclude that the gravitational force
near the surface of the earth is always
downward toward the center of the
earth and of has the magnitude
Fgravitational = mg
where g is a constant = 9.8m/s2
Summary:
The gravitational force on an object is
always proportional to its mass.
near the surface of the earth, the acceleration
due to the gravitational force is constant
so that
Fgravitational =mg (downward)
g is approximately 9.8m/s2 (depends on
altitude)
Constant acceleration.
In some kinds of motion, including
free fall of an object in the gravitational
field of the earth, the instantaneous
acceleration of a moving object remains
the same over some time interval.
In that case, the average acceleration is
the same as the instantaneous acceleration,
a plot of speed versus time is a straight line,
and the average speed is ½ the sum of
the initial and the final speed.
If toss a ball up in the air, what is the
direction of the velocity,
the acceleration and the total force on the
ball just before it leaves my hand?
v
a
F
A.
u
u
u
B.
u
d
d
C.
u
d
u
D.
u
u
d
If I toss a ball up in the air, what is the
direction of the velocity,
the acceleration and the total force on the
ball just after it leaves my hand?
v
a
F
A.
u
u
u
B.
u
d
d
C.
u
d
u
D.
u
u
d
Answer B.
Velocity is up.
The only force acting is gravity, down.
The acceleration is in the same direction as
the total force.
If I toss a ball up in the air, what is the
direction of the velocity,
the acceleration and the total force on the
ball at the top of the trajectory?
v
a
F
A.
d
u
u
B.
0
d
u
C.
0
d
d
D.
u
0
0
E.
0
0
0
Answer C.
Velocity is 0
Force is still down (gravitational).
Therefore acceleration is also down.
To see this another way, note that just before this
instant velocity is up, and just after it is down,
so velocity is changing.
If I toss a ball up in the air, what is the
direction of the velocity,
the acceleration and the total force on the ball
just before it hits the floor?
v
a
F
A.
d
u
u
B.
d
d
d
C.
d
u
u
D.
d
0
0
E.
0
0
0
Answer B:
Velocity is now down.
Total force and acceleration remain down
until it hits.
Electromagnetic forces
Essentially all the forces which we encounter
in everyday life are either gravitational or have
their origin in what physicists call electromagnetic
forces. We can introduce them by thinking
about holding an object up so that it doesn’t fall.
The force arising from my hand on the book
is of electromagnetic origin. This is basically
because some electrically charged particles
in my hand (called electrons) repel some
electrically charged particles in the book
(also electrons). Notice that in this case
the force is of SHORT RANGE. That is,
it doesn’t act unless my hand actually touches
the book. That’s very different from the
gravitational force. The earth pulls gravitationally
on the book even if the earth doesn’t touch the
book.
As I hold the book, what is the magnitude
and direction of the force of electromagnetic
origin that I exert on the book, assuming
I hold it still for a long time. The mass of
the book is M.
A
B.
C.
D.
Mg down
Mg up
0
can’t tell with this information
Answer B:
The acceleration is zero. Therefore the
total force is zero.
The total force is the sum of the gravitational
force Mg down and the force of electromagnetic
origin from my hand Fhand:
Ma=0=Fhand-Mg
So Fhand=Mg (positive sign means it’s up)
Note the force has magnitude Mg but it’s of
electromagnetic origin.
An important aspect of this example of
the book is that the forces of different types
(here gravitational and electromagnetic) added
together to make the total force. Because they
were in opposite directions, the two forces
added to zero (cancelled out) to make a total
force of zero.
Cart on a track.
What is the direction of the acceleration
of the cart?
A.
B.
C.
D.
straight down
Downward along the track
Zero
Can’t tell without measuring
Answer B..
Here the track was tilted up and the
acceleration of the cart was downward along
the track (not straight down)
Which of the following is true of
the electromagnetic force on the cart?
(neglect friction)
A It's down along the track and smaller than Mg
B. It's upward along the track and smaller than Mg
C. It's upward and equal to Mg
D It's upward and smaller than Mg
E. It's perpendicular to the track upward and
less than Mg.
Answer E:
The downward arrow shows the gravitational
force. The acceleration is along the track downward, so the total force must also be downward
along the track (small arrow parallel to the
track) Therefore the electromagnetic force
must cancel the rest of the gravitational force
and does it by pointing up and perpendicular
to the track, as shown. (Here we ignore friction.
With friction the electromagnetic force will point
a little to the left of the way it is shown.)
Summary
From position and time data we get
velocity and acceleration.
From acceleration and mass we
define the total force on an object F=ma.
Two kinds of forces, gravitational, electromagnetic:
Gravitational are always proportional to the mass .
Near earth's surface, gravitational force is mg
down.
Electromagnetic forces are often of finite range.
NOT proportional to mass.
Different kinds of forces can add to give the
total force and may cancel each other out.