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Re: [Phys-l] Separating inertial mass and g mass. Was: Re: adifferent kind of math background quiz



I take it that horologists use an "effective inertial mass" rather than the
physics definition of inertial mass, to bundle the effect of the surrounding
fluid into the mass. Or do they use and "effective gravitational mass"?

John M. Clement
Houston, TX

They must do both, as the air has two effects on the bob. The bob drags some air on its surface and pushes air; both increase its inertia. OTOH the flotation effect reduces the restoring force, i.e. equivalent to reducing its g. mass. As a rule of hand (thumb is too fine) the former effect is about twice the latter. The former, of course, is highly dependent on its shape, size of the enclosing case, etc.

I asked my question in order to separate the buoyancy effect from the damping effect. I have since re-discovered that the damping effect is very small -- overwhelmed by the circular error in my example. Amplitude 0.1 => 0.3 (radian) results in a change in frequency of ~ 6/10k, while a change in the damping coefficient [the c in exp(-c*t)] of 0.005 => 0.05 is 5/100k. [Q: ~ 400 and 40, respectively. period ~ 1.5 s]

bc still wants somene to write the diff. eq. to include the buoyancy


On 2010, Jan 09, , at 11:37, John Clement wrote:

Gravitational mass and inertial mass are the same only in the
gravitational field alone. All forces other than gravity tend to mask this
equivalence. One of the reasons of discrepancy with horological
corrections for a physical pendulum might be the dependence of the forces
involved on shape of moving object (a drag force on the swinging bob
depends on its velocity AND on its size, and even on degree of smoothness
of its surface if we go for high accuracy). An open parachute is a good
illustration. For a class demo I used sometimes a box with chalk and a
sheet of paper dropped together, and in the second trial did the same with
the same sheet thoroughly crumpled so that now both items hit the floor
simultaneously. By changing the shape of the sheet we could uncover the
equivalence of the inertial and gravitational mass. My personal feeling is
that the students were impressed by this simple demo.
But the equivalence between the gravitational and inertial masses is
actually very subtle, and it is doubtful that the simple demo impressed this
on the students. If they focus on anything, they will just say of course,
when you lower the air resistance the two fall at the same rate. I suspect
if you have them predict the results, they will predict the correct results.
Actually without a prediction, they will not remember the demo, and if it is
an obvious result, then it will not have much effect on their thinking.


You suggest that counter intuitive demos. have a more lasting effect?

Part of the problem with establishing this is that students always see
g=9.8m/s^2 used to calculate the gravitational force. So immediately the
distinction between the gravitational and inertial forces has been assumed.
But if g=9.8 N/kg where F_g = mg, then the inertial and gravitational forces
are not assumed to be the same. There is also the difficulty with this
equation that students find it difficult to understand because how can you
use an acceleration find the force on something that is not accelerating.

To bring up the issue of gravitation vs inertial mass one must establish
that they are initially different things and have students understand this
comes from 2 different equations. Since even students in an intro calculus
based course often do not really understand variables or equations, this is
a tall order. They sometimes know how to plug numbers into equations, but
understanding that equations are descriptions of relationships, and can have
meaning is something that the math teachers have been remiss in getting them
to comprehend.

I take it that horologists use an "effective inertial mass" rather than the
physics definition of inertial mass, to bundle the effect of the surrounding
fluid into the mass. Or do they use and "effective gravitational mass"?

John M. Clement
Houston, TX

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