# Rate of acceleration for two hanging masses in two experiments

I conducted an experiment where two masses of different weight were attached by a string and fed through a pulley. The difference between the masses was $0.025\: \mathrm{kg}$. The lighter mass was pulled to the the floor, causing the heavier mass to be suspended in the air. We timed how long it took for the heavy mass to reach the ground once the lighter weight was released. The average acceleration of the system was $0.883\: \mathrm{m/s^2}$.

We then performed the exact same procedure but decreased the total weight of the system. The difference between the masses was still $0.025\: \mathrm{kg}$. The average acceleration for this system was $0.922\: \mathrm{m/s^2}$.

I am curious as to why the rate of acceleration (both in theory and in practice) is slower for the heavier set of experiments, and is faster for the lighter set of experiment, even though the difference between the masses is the same.

• Hint. Nov 16 '14 at 16:29

One way to think of this conceptually is that the inertial mass of the system is in fact different from the gravitational mass. That is, while the net force on the systems are the same, the two systems have a different amount of mass, and so they resist changes in velocity to different degrees. Except that in this case, we are considering a compound system, not just a single object.

In your first case, the mass difference, and thus the effective gravitational mass of the system, was $0.025\:\rm kg$. If we multiply this by the gravity, we can get the net force on the system. $$0.025\:\rm kg \times 9.81\: m/s^2 \approx 0.2452\: N$$

This caused the system to accelerate at $0.883\: \rm m/s^2$. Now, using Newton's second law, we can find the inertial mass of the system. $$m = \frac{F}{a} = \rm \frac{0.2452\: N}{0.883\: m/s^2}\approx 0.2777\: kg$$

So, this system has a total mass of about $\rm 0.2777\: kg$. We could also find the masses of the individual weights, since we know their sum and difference; this I leave as an exercise to the reader.

The second case can be solved the same way. The mass difference and thus net force are the same in this case, the only thing that changed was the acceleration. Thus: $$m = \frac{F}{a} = \rm \frac{0.2452\: N}{0.922\: m/s^2}\approx 0.2660\: kg$$

As can be seen, the total mass in the first case was $\rm 0.2777\: kg$, which is indeed larger than the second case, which has only $\rm 0.2660\: kg$.