Is the acceleration due to a fictitious force independent of mass in general?
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Answered by
0
Intuitively (at least to me) it seems that the answer should be "yes", since a fictitious force arises due to being in a non-inertial frame; the frame is accelerating, but the objects within this frame (neglecting any other forces) and so they will remain at rest (or at constant velocity), however, relative to an observer in this non-inertial frame, they will all seem to accelerate in the opposite direction at the same rate.
If the above is correct, then this notion can be shown mathematically as follows (at least, I think). Newton's 2nd law states that the acceleration, a
a
of a massive object (due to an applied force) is proportional to the force, F
F
acting on it, such that F=ma
F
=
m
a
, where the mass m
m
of the object is the constant of proportionality. This does not define the force (by which I mean it doesn't give a mathematical expression describing the force, for example, the Coulomb force is defined by the mathematical expression FC=14πε0q1q2r2r̂
F
C
=
1
4
π
ε
0
q
1
q
2
r
2
r
^
which is equal to ma
m
a
). However, in the case of a fictitious force, Ffic
F
f
i
c
, the force is defined by the mathematical expression Ffic=−mainert
F
f
i
c
=
−
m
a
i
n
e
r
t
, where ainert
a
i
n
e
r
t
is the acceleration of the reference frame. It has this form since it is introduced to account for the acceleration of the reference frame. Then, by Newton's 2nd law, assuming that no other forces are acting, we have that
−mainert=ma⇒a=ainert
−
m
a
i
n
e
r
t
=
m
a
⇒
a
=
a
i
n
e
r
t
i.e. the acceleration of each of the objects due to this fictitious force is independent of its mass.
I'm unsure though whether I've conducted this analysis completely correctly though, so any feedback would be much appreciated.
If the above is correct, then this notion can be shown mathematically as follows (at least, I think). Newton's 2nd law states that the acceleration, a
a
of a massive object (due to an applied force) is proportional to the force, F
F
acting on it, such that F=ma
F
=
m
a
, where the mass m
m
of the object is the constant of proportionality. This does not define the force (by which I mean it doesn't give a mathematical expression describing the force, for example, the Coulomb force is defined by the mathematical expression FC=14πε0q1q2r2r̂
F
C
=
1
4
π
ε
0
q
1
q
2
r
2
r
^
which is equal to ma
m
a
). However, in the case of a fictitious force, Ffic
F
f
i
c
, the force is defined by the mathematical expression Ffic=−mainert
F
f
i
c
=
−
m
a
i
n
e
r
t
, where ainert
a
i
n
e
r
t
is the acceleration of the reference frame. It has this form since it is introduced to account for the acceleration of the reference frame. Then, by Newton's 2nd law, assuming that no other forces are acting, we have that
−mainert=ma⇒a=ainert
−
m
a
i
n
e
r
t
=
m
a
⇒
a
=
a
i
n
e
r
t
i.e. the acceleration of each of the objects due to this fictitious force is independent of its mass.
I'm unsure though whether I've conducted this analysis completely correctly though, so any feedback would be much appreciated.
Answered by
0
Acceleration due to gravity is independent of mass. Force due to gravity is dependent on themass of the two objects x the universal gravity constant divided by the distance.
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