Inertial and Gravitational mass - Assertion Reason

[Jahnavi]

Homework Statement

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Homework Equations

The Attempt at a Solution

I think the two masses , inertial and gravitational are equivalent and because of this fact , the ratio of the two masses is one . The correct option should be 1) .

But the answer given is 3) i.e A statement is true and R statement is false .

I would request experts to give their opinion .

 

[Bandersnatch]

I think the two masses , inertial and gravitational are equivalent and because of this fact

What is the reasoning you would use to show that they are indeed equivalent?

 

[Jahnavi]

What is the reasoning you would use to show that they are indeed equivalent?

I think this has been found experimentally .

Also , this is something which is written in the books ( not my textbook ) and also at several places on the web .

But I might have misunderstood things .

 

[Bandersnatch]

If you were write down the equation of motion for a point mass in a gravitational field, it'll look like this:
$$m_i a = G\frac{Mm_g}{r^2}$$

The point is, at least as I understand the intent of the question, that there is no theoretical Reason - in Newtonian physics - for the two masses (##m_g## and ##m_i##) to be equal. This would be better visible if we used completely different symbols for the two masses. Say, ##i## for the interial mass.
But they appear to be equal, to the best ability of our experimental measurements, so we Assert that they're equal. I.e. we treat them as equal, which handily simplifies our equations (you can now cancel out these two masses in the above equation).
This is not a reason for their equality, just a useful assertion. It could in principle so happen, that an experiment tomorrow finds that the two are, after all, different after however many decimal places - this would show that our assertion was wrong.

 

[Jahnavi]

Don't you think if the latter statement R is not correct then this invalidates the former statement A ?

I mean if the two are not equivalent , then their ratio cannot be one and option 4) would be correct .

 

[Bandersnatch]

If you have a reason for something being true, then you don't have to assert it being so.
If you don't have a reason, then you can assert the truth. Your assertion does not provide a reason for the veracity of the statement (that would be circular reasoning).

 

[Jahnavi]

Sorry , I couldn't understand

 

[Bandersnatch]

Sorry, let me try again.

The A statement says that they have the same value.

The R statement says that they're the same thing, conceptually.

We know A to be true to the best ability of our observations.

We don't have a reason to think that R is true. In the equation provided above these are two different variables.

So R cannot support A. If these were the same variable, then this fact would support their equality.

 

[Jahnavi]

I think I have not been able to distinguish properly between words 'equal' and 'equivalent' .

Are you saying that the two types of masses have been observed to have same value experimentally but conceptually these are two different things ?

Can the two statement be read together as => Despite the two masses being different things conceptually(theoretically different) , they have been found to have same value experimentally ?

 

[Bandersnatch]

I believe that's a good representation of the question's intent, yes.

 

[Jahnavi]

Thank you very much

 

[Jahnavi]

@Bandersnatch please see another similar problem .

The answer given is 2) .

Do you agree it is wrong and the correct answer should be 3) just like in the OP ?

Would you agree that if the word 'equivalent' is replaced by 'equal' then correct option would be 2) ?

 

[Bandersnatch]

In the light of the second question I've no idea what they mean.
Can you provide more context for the whole questionnaire? What is it testing? Was there some additional information given? Is this one in a different section of the test than the one before?

 

[Jahnavi]

Can you provide more context for the whole questionnaire?

This is a complete question . It is a similar Assertion Reason problem just like the one in OP

'A' statement is the Assertion and 'R' statement is the Reasoning .

The answer has to be marked as per the instructions in the first picture in the OP .

Is this one in a different section of the test than the one before?

Yes . The second one is from a different book . It is in the back end exercise of chapter Gravitation .

 

[haruspex]

I think it helps to clarify first what we mean by inertial and gravitational mass.
Assume we have a way of measuring force, e.g. by spring compression. In a given gravitational field, we can thus find the weight of an object. Independently, we can find what acceleration is produced by subjecting the object to a known force, and define its inertial mass as F/a.
Thus we can find the weight (in a given field) and inertial mass of the same object.
Next, we observe that across different objects these are in a constant ratio, and we are therefore able to characterise the gravitational field strength as such-and-such an acceleration.
Having done that, we can define the gravitational mass as F_g/g; it follows that the two mass determinations are equivalent.

The observation that all bodies fall at the same acceleration starts, as we all know, with Galileo, but is also the cornerstone of General Relativity. Thus I would venture that the R proposition here is true, and that the correct answer to post #1 is (1).

Note that the given answer (2) to post #12 claims that this same R clause is true, thereby contradicting the book answer to the post #1 problem.

So what about R leading to A in post #12? First, what does it mean to say that the man experiences no gravity?
Each day we experience gravity by the fact that the ground has to exert a force upwards on us for us to stay in the 'same' place. The man's failure to experience gravity consists of his staying in the same place (within the cabin) yet feel no applied force. This happens because he and the cabin accelerate equally in the same gravitational field. It follows that this is equivalent to Galileo's observation.
As I showed above, that is considered explained by the equivalence principle. So, again, I would mark (1) as correct.