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This question already has an answer here:

Why does a larger mass in a pendulum have the same period as a lighter mass? i know it has something to do with gravity and length but how can this be explained in depth? like for example the galileo's experiment where both masses were nearly the same but the lighter mass was slower (air resistance)

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marked as duplicate by twistor59, Qmechanic Jul 28 '13 at 19:37

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  • $\begingroup$ Same reason as why an elepphant, a donkey, a human, an ant, and an apple, and an a bacteria fall at the same reason . $\endgroup$ – Abhimanyu Pallavi Sudhir Jul 28 '13 at 9:55
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    $\begingroup$ EXACT duplicate of Why doesn't mass of bob affect time period? $\endgroup$ – udiboy1209 Jul 28 '13 at 11:31
  • $\begingroup$ When you analyze $\tau = I\alpha$ for a pendulum undergoing small oscillations, you will find that the mass terms cancel on both sides and hence the period does not depend on mass of the pendulum. $\endgroup$ – Antillar Maximus Jul 28 '13 at 13:11
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The reason lies in the nature of gravity: under the influence of gravity, all bodies, no matter what their mass is, accelerate at the same rate. This is also true for bodies attached to a rope, resulting in a pendulum. The force driving the pendulum is gravity, and hence, its rate of acceleration (which effectively determines the period) is the same for all masses.

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When you write Newton's equations of motion to determine how the mass $m$ moves, you get the following $$ F = ma $$

It should be vectorial, but that's irrelevant for the point I'm making. So in order to know how the mass moves, you need to determine its acceleration $a$. The missing thing to do that is the force $F$, which will depend on the situation. In this one, $F$, the force of gravity, is proportional to the mass $m$. So the mass drops out from both sides of the equation. In other terms, in the situation of the pendulum, the acceleration of the object will depend on several things, but not on the mass.

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