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Thursday, March 19, 2009

APPLICATIONS OF NEWTON’S LAWS – I

1. An obvious conclusion from is that if 0 then 0 ! How simple, yet
how powerful ! This says that for any body that is not accelerating the
must va
F ma F a
sum of all
the forces acting upon it
= = =
nish.
2. Examples of systems in equilibrium: a stone resting on the ground; a pencil balanced
on your finger; a ladder placed against the wall, an aircraft flying at a constant speed
and constant height.
3. Examples of systems out of equilibrium: a stone thrown upwards that is at its highest
point; a plane diving downwards; a car at rest whose driver has just stepped on the
car's accelerator.
4. If you know the acceleration of a body, it is easy to find the force that causes it to
accelerate. Example: An aircraft of mass m has position vector,
r􀁇 3 2 4
2 2
2 2
2
( )ˆ ( )ˆ
What force is acting upon it?
SOLUTION: ˆ ˆ
6 ˆ (2 12 )ˆ
5. The other way around is
at bt i ct dt j
F md xi md y j
dt dt
bmt i m c d t j
= + + +
= +
= + +
􀁇
2
2
not so simple: suppose that you know and you want to
find . For this you must solve the equation,
This may or may not be easy, depending upon
F
x
d x F
dt m
=
(which may depend upon both as
well as if the force is not constant).
6. Ropes are useful because you can pull from a distance to change the direction of a
force. The tension, often denote
F x
t
d by , is the force you would feel if you cut the
rope and grabbed the ends. For a massless rope (which may be a very good
approximation in many situations) the tension is the same at every p
T
oint along the
rope. Why? Because if you take any small slice of the rope it weighs nothing (or
very little). So if the force on one side of the slice was any different from the force
force on the other side, it would be accelerating hugely. All this was for the "ideal
rope" which has no mass and never breaks. But this idealization if often good enough.
7. We are all familiar with frictional force. When two bodies rub against each other,
the frictional force acts upon each body separately opposite to its direction of motion
(i.e it acts to slow down the motion). The harder you press two bodies against each
other, the greater the friction. Mathematically, , where is the force with which
you press the two bodies against ea
F= μN N
􀁇 􀁇 􀁇
ch other (normal force). The quantity is called the
coefficient of friction (obviously!). It is large for rough surfaces, and small for smooth
ones. Remember that F N is an empirical re
μ
= μ
􀁇 􀁇
lation and holds only approximately.
This is obviously true: if you put a large enough mass on a table, the table will start to
bend and will eventually break.
8. Friction is caused by roughness at a microscopic level
- if you look at any surface with a powerful microscope
you will see unevenness and jaggedness. If these big bumps
are levelled somehow, friction will still not disappear because
there will still be little bumps due to atoms. More precisely,
atoms from the two bodies will interact each other because of the electrostatic interaction
between their charges. Even if an atom is neutral, it can still exchange electrons and there
will be a force because of surrounding atoms.
9. Consider the two blocks below on a frictionless surface:
1 2
2
1
We want to find the tension and acceleration: The total force on the first mass is
and so . The force on the second mass is simply and so . Solving the
above, we get:
F T
F T ma T T ma
T m F
m m

− = =
=
+ 2 1 2
and a F .
m m
=
+
10. There is a general principle by which you solve equilibrium problems. For equilibrium,
the sum of forces in every direction must vanish. So 0. You may always
choose the , ,
x y z F F F
x y z
= = =
directions according to your convenience. So, for example, as in the
lecture problem dealing with a body sliding down an inclined plane, you can choose the
directions to be along and perpendicular to the surface of the plane.
1 m 2 F T T m

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