Are the Reasons for Pendulum's Mass and Amplitude Independence Contradictory?

[Kaneki123]

Okay...The time period of a pendulum is independent of its mass and amplitude.For the "independence of mass",the reason is that, for same amplitude (distance from mean position), any change in mass would have no effect on the acceleration of the mass (gravitational acceleration is constant);hence the mass will reach the mean position at same time...But for the "independence of amplitude'', the reason I found is that, for a same mass, increase in amplitude would result in increase in restoring force, which would result in increase in acceleration; increased distance would increase acceleration and hence the mass will reach the mean position at the same time...Now my question is that, are'nt these two reasons contradictory, like one says that acceleration remains constant and the other states that the acceleration would increase?Any help will be appreciated...

 

[Kaneki123]

Okay...Any freely falling body on Earth will have acceleration of 9.8 ms^-2.Suppose that in case of a pendulum, the bob is taken to the right of its mean position...Now only a ''component of weight'' will be acting as a net force on the bob...My question is that, will the acceleration of the bob in this case also be 9.8ms^-2 (same as gravitational acceleration), or will it be different?

 

[sophiecentaur]

Okay...The time period of a pendulum is independent of its mass and amplitude.

Well, it isn't exactly true for a pendulum. Simple Harmonic Motion is only followed when the restoring force is exactly proportional to displacement - as with a steel clock spring.
But there is no way to discuss the details of the motion without using Maths and solving the differential equation of motion. That's the only real explanation and arm waving cannot suffice, I'm afraid.
The acceleration towards the rest position is not constant but is proportional to the displacement. You have more or less described what goes on but what you can't 'prove' that way is how that effect produces equal time periods of cycles of all amplitudes. That has to use the Calculus.

 

[sophiecentaur]

Okay...Any freely falling body on Earth will have acceleration of 9.8 ms^-2.Suppose that in case of a pendulum, the bob is taken to the right of its mean position...Now only a ''component of weight'' will be acting as a net force on the bob...My question is that, will the acceleration of the bob in this case also be 9.8ms^-2 (same as gravitational acceleration), or will it be different?

With the bob at the same level as the pivot, the downwards acceleration will be g. Not for other angles, though. At other angles, the force towards the pivot becomes relevant and the motion is due to mg and the tension. This link says it all, with pictures.

 

[sophiecentaur]

I think you may have Mod trouble for posting about the same topic on two threads. You will probably be 'merged'!

 

[fresh_42]

Two threads have been merged as the issue could possibly be solved for both. This is an exception, because both questions are closely related.

 

[Kaneki123]

Well, it isn't exactly true for a pendulum. Simple Harmonic Motion is only followed when the restoring force is exactly proportional to displacement - as with a steel clock spring.
But there is no way to discuss the details of the motion without using Maths and solving the differential equation of motion. That's the only real explanation and arm waving cannot suffice, I'm afraid.
The acceleration towards the rest position is not constant but is proportional to the displacement. You have more or less described what goes on but what you can't 'prove' that way is how that effect produces equal time periods of cycles of all amplitudes. That has to use the Calculus.

Can you give me a valid reason as to why the time period does not (or a little) depend upon the mass of the bob?

 

[rumborak]

That aspect I feel is a bit more obvious. I mean, disregarding friction, all objects accelerate at the same rate towards earth, independent of mass. So, imagine the pendulum is at a horizontal 90 degree position (extreme case, but just for illustration). If you split the weight into two parts that are now hung off two separate connections, the weights will fall at the same rate as the big combined weight.

 

[sophiecentaur]

Can you give me a valid reason as to why the time period does not (or a little) depend upon the mass of the bob?

Can you give me a "valid reason" why you didn't read the link I gave you? It has all you need to know about a simple pendulum, including how the mass disappears from the derived equation. We do expect people to try, just a little and not be spoon fed with everything.

 

[Dr.D]

Is there some reason why math is being avoided here? It seem like a lot of this could be address quite simply with the relevant diagram and associated differential equation of motion.

 

[Jimmy]

Is there some reason why math is being avoided here? It seem like a lot of this could be address quite simply with the relevant diagram and associated differential equation of motion.

See the link in post #4 of this thread.

 

[Kaneki123]

That aspect I feel is a bit more obvious. I mean, disregarding friction, all objects accelerate at the same rate towards earth, independent of mass. So, imagine the pendulum is at a horizontal 90 degree position (extreme case, but just for illustration). If you split the weight into two parts that are now hung off two separate connections, the weights will fall at the same rate as the big combined weight.

the reason you gave brings me to my original question that, mass is excluded because gravitational acceleration is constant(regardless of distance from mean position), but in case of amplitude, the amplitude is excluded because acceleration is directly proportional to distance from mean position?

 

[Nugatory]

but in case of amplitude, the amplitude is excluded because acceleration is directly proportional to distance from mean position?

That is the defining property of a harmonic oscillator: The restoring force and hence the acceleration is directly proportional to the displacement. That condition leads to a differential equation in which the amplitude drops out.

Ir is worth noting that a pendulum is not a perfect harmonic oscillator, but as long as the angle of swing is small enough that $\sin\theta\approx\theta$ and the mass is concentrated in a point at the end of the string it's pretty close.

 

[sophiecentaur]

the reason you gave brings me to my original question that, mass is excluded because gravitational acceleration is constant(regardless of distance from mean position), but in case of amplitude, the amplitude is excluded because acceleration is directly proportional to distance from mean position?

Several hundred years ago it was acknowledged that most aspects of Science - particularly Physics - are described best using mathematics. It is, as far as I can see, futile to try to produce alternative, hand waving explanations when the explanation is perfectly clear to anyone who is prepared to do the Maths. People who are not prepared to involve the Maths must really just accept simpler explanations because they are not in a position to argue against them.
The Maths in the link I posted way back, answers all your questions.

 

[person123]

The force of gravity acts down while the force which accelerates the pendulum is tangential to its circular motion. When the amplitude of the pendulum increases, the direction of these two forces gets more similar which means that a larger component of the force of gravity is rotating the pendulum. This force is F_gsinθ with θ being the amplitude. This is clearly shown in a image from the link.

Note that the force of gravity itself remains constant.