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Ch. 6 FORCE AND MOTION  II
6.1 Newton’s Law in Non-inertial Reference Frames
6.1.1
Inertial force in linear acceleration reference frame
From the view of the ground:


f  ma
where the spring acting on the
ball provides the force for the
forward acceleration motion.

a
However, in the view of an observer rest at the acceleration
vehicle, the ball exerted by a spring force but keep in stationary.
To explain this phenomenon, an inertial force is introduced into
its force diagram:

f

f *
(f = kx)
We have
 
f  f*0
Therefore,



f *   f  (ma )


f *  ma
6.1.2
Inertial force of centrifugation
In natural coordinates, a ball rotating
with the plate sustaining a centripetal
force:
v2
 m 2 r
fn  m
r

However, in the view of an observer rest at the rotating plate,
the ball exerted by a tensile force but kept in stationary. To
explain this phenomenon, an inertial force of centrifugation is
introduced into its force diagram:

f

f*
(f = m r)
2
Therefore,
 
f  f*0


2

m

rrˆ
f* f

2
f *  m r
6.1.3
Coriolic’s force
In the view of ground, the mass
m, which is moving along the
radius direction, sustains a force by
the string perpendicular to the
radius direction. This force produce
a transverse acceleration.


v'
B'
A'
(B'')
O
A

B

s  BB' AA'  (OB  OA)t  ABt
1
2
 (v' t )t  v' (t )  aC t 2
2
F  ma C  2mv ' 
However, in the view of an observer rest at the rotating plate,
the ball exerted by a force in transverse direction but without a
motion in this direction. To explain this phenomenon, a Coriolic’s
force is introduced:
*

f C  F
This imaginary force is called Coriolic’s force. Its magnitude is
f C*  2mv'
Considering together with the direction, it can be expressed in
terms of cross product as
*
 
f C  2mv '
6.2 Galilean Relativity
6.2.1
Galilean transformations
    
r  r 'R  r 'Vt
y'
y
t'  t

V
x
z
z'
x'
 x'  x  Vt

 y'  y

 z'  z
 t '  t
6.2.2
Galilean relativity
All the inertial reference frames are equivalent. Acceleration is
invariant to the choice of inertial system:
 
a  a'
Generalizing, we assert that there exists a system of mechanics,
based on Newton’s laws of motion and called Newtonian
mechanics, that describes particle motion in our everyday work to
high accuracy. A principle of relativity goes with it: Galilean
relativity.
For Newtonian mechanics, there is no preferred inertial
reference system.
Problems:
1. 6-41 (on page 115)
2. 6-42
3. A ramp with a mass of M in on a frictionless table. A block
has a mass of m slides down from the ramp (see figure 1).
There is no friction between the block and the ramp. What
are their accelerations? What is the acceleration of m relative
to M?
m
M

FIGURE 1
4. Assume that the Earth is a sphere and that the force of gravity
(mg) points precisely toward the center of the Earth. Taking
into account the rotation of the earth about its axis, calculate
the angle between the direction of a plumb line and the
direction of the Earth’s radius as a function of latitude. What
is this deviation at a latitude of 45.
5. A mass is attached to the lower end of a string of length l; the
upper end of the string is held fixed. Suppose that the string
initially makes an angle  with the vertical. With what
horizontal velocity must we launch the mass so that it
continues to travel at constant speed along a horizontal
circular path under the influence of the combined forces of
the tension of the string and gravity? This device is called a
conical pendulum.