Relationship between mass and acceleration

[Anthony Lu]

Hi. I am currently working on an assessed science experiment write-up. The experiment is about whether or not if mass of a pendulum bob affect the oscillation time for a simple pendulum. Obviously, the answer is no but I am struggling to write a concise and accurate explanation to why this is. I believe it relates to gravity, acceleration and all that stuff that relates to Newton's second law but I'm not sure if what I have written is accurate. If someone would be so kind and check the accuracy of what I wrote, the below is what I currently have for my hypothesis, explanation to hypothesis, conclusion and explanation to conclusion and the science behind why mass does not affect oscillation time.

Hypothesis:The mass of the pendulum bob does not affect the oscillation time of a simple pendulum. This is because the force acting on the pendulum bob causing its movement is the gravitational force of earth, which stays the same at 10m/s^2 (or 9.8m/s^2). Hence, the acceleration of the pendulum bob will stay at 10m/s^2 no matter the mass. This is because according to Newton’s 2nd law, F=ma. When the mass of the object is changed, for example doubled, the net force of the object doubles. Because a=F/m, changing the mass will not affect acceleration as the doubled net force and doubled mass can cancel out (F2/m2).Conclusion:According to my results table and the graph, my hypothesis was correct. No matter the mass of the pendulum bob, the oscillation time does not change. This was proven through the graph, where a straight line of best fit was produced to show little to no change in time needed for a simple pendulum to complete one oscillation.The mass does not affect the oscillation time of the pendulum bob because the bob is being accelerated towards the ground at a constant rate of 10m/s^2. Just as objects with different masses but similar shapes fall at a same rate, the pendulum is pulled downward at the same rate no matter the mass of the bob, hence the bob will take an equal amount of time to complete one oscillation, assuming there is no air resistance.This can be further proven by Newton’s second law, as explained in the hypothesis. When the mass of the object is doubled, the net force of the object doubles because Fnet = ma. When a= F/m, the doubled net force and doubled mass cancels out, and the acceleration ends up the same. This proves why the gravitational force/acceleration of 10m/s^2 never changes and hence, it proves why the time it takes for a simple pendulum to complete one oscillation stays the same, even when the mass of the pendulum bob is changed.

 

[Dale]

the gravitational force of earth, which stays the same at 10m/s^2 (or 9.8m/s^2)

I would say “the gravitational force of the earth, which is proportional to the mass, so the acceleration stays the same.”

I think the rest of the paragraph shows that is what you intended and that you are thinking about it correctly.

 

[Anthony Lu]

I would say “the gravitational force of the earth, which is proportional to the mass, so the acceleration stays the same.”

I think the rest of the paragraph shows that is what you intended and that you are thinking about it correctly.

Thanks but wouldn't what you said mean that if the mass increases, the gravitational force increases and hence the acceleration increases? If so isn't it opposite to what I'm conveying in my conclusion? Sorry I am not quite sure but thanks.

 

[Dale]

wouldn't what you said mean that if the mass increases, the gravitational force increases

Yes.

and hence the acceleration increases?

No. Remember Newton’s 2nd law. The force increases, but so does the amount of force required for a given acceleration. The two effects cancel out.

 

[Anthony Lu]

Yes.

No. Remember Newton’s 2nd law. The force increases, but so does the amount of force required for a given acceleration. The two effects cancel out.

Oh so basically what you're saying is similar to like when a = F/m, where the mass is doubled, the force will be doubled and hence they will cancel out so the acceleration remains the same and from what you said, m = 2F/2g (Oh and another question : Is there a scientific theory behind why force doubles when the mass is doubled?)

 

[Dale]

Oh so basically what you're saying is similar to like when a = F/m, where the mass is doubled, the force will be doubled and hence they will cancel out so the acceleration remains the same

Yes, which is what you wrote later on anyway.

Is there a scientific theory behind why force doubles when the mass is doubled?)

Not really, it is basically just an observation. We observe that the acceleration is independent of the mass. The kind of force law that produces an acceleration independent of mass is one that is proportional to mass. So we propose the theory ##f=GMm/r^2## to match the observations.

In general relativity this observation is embedded even more deeply in the theory, but it works in a way that gets away from the idea of the force of gravity somewhat.

 

[Anthony Lu]

Ah okay thanks a lot! Just a dumb question here. So the force in F=ma when considering a free-falling object is weight right?

 

[Dale]

So the force in F=ma when considering a free-falling object is weight right?

Yes, exactly

 

[Anthony Lu]

Yes, exactly

Right. So...

The mass of the pendulum bob does not affect the oscillation time of a simple pendulum. This is because the force due to gravity acting on the pendulum bob causing its movement (downward acceleration of falling object) is proportional to the mass. Hence, the acceleration of the pendulum bob will stay at 10m/s^2 no matter the mass. This is also according to Newton’s 2nd law, F=ma. When the mass of the object is changed, for example doubled, the force of the object (weight) doubles as it requires double the amount of force to move the object. Because a=F/m, changing the mass will not affect acceleration as the doubled net force and doubled mass can cancel out (F2/m2). Because acceleration does not change, the oscillation time of a simple pendulum never changes no matter how much the mass is changed.

 

[Dale]

Looks good to me!

If you really want to go the extra mile then you could show that the tension in the pendulum arm is also proportional to the mass. That is not as easy as for gravity, but it can be done.