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**Physics, Astronomy & Cosmology / Re: what would happen if gravitational mass were different than inertial mass? **

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**Yesterday**at 10:15:05 »

Quote from: Colin2B

similarly with the pendulum having different masses, you would get different periodsI think most models of unequal gravitational & inertial mass would still assume that they were proportional to each other.

- Otherwise you could end up with a situation where a zero gravitational mass had a non-zero inertial mass (or vice-versa)

- Or (more extreme), a positive gravitational mass had a

*negative*inertial mass (or vice-versa)

A pendulum works by the periodic interchange of gravitational potential energy (calculated from gravitational mass) & kinetic energy (calculated from inertial mass).

- The key being that the mass of the pendulum cancels out in the calculation

- so two pendulums with different masses (but same length & gravity field) have the same period.

Lets say m

_{i}= k m

_{g}

- m

_{i}is inertial mass

- m

_{g}is gravitational mass

- And k is some constant, close to 1

From the "Energy" method of calculating the period of a pendulum:

m

_{g}gh = 1/2 m

_{i}v

^{2}

or km

_{i}gh = 1/2 m

_{i}v

^{2}

- Where g is the acceleration due to gravity (assumed uniform)

- h is the height of the pendulum

- v is the velocity of the pendulum

Change in velocity due to a change in height is:

v=SQRT(2kgh) (note that m

_{i}still cancels out)

So the period would be slightly

*different*than if m

_{i}= m

_{g}...

- But the period is still independent of the mass of the pendulum.

See: https://en.wikipedia.org/wiki/Pendulum_(mathematics)#Simple_gravity_pendulum

..and expand the section on the "Energy derivation".

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