Forces on a Book: Scale Reads 3 & 6 - How Can This Be?

[Mechanic]

1.1 A weight scale is placed on a table on Earth, and a book of mass m is placed on the scale.
1.2 A force of magnitude mg is exerted on the book.
1.3 An equal and opposite force is exerted on the book by the table.
1.4 The book does not accelerate because there are equal and opposite forces of magnitude mg acting on it.
1.5 The scale is calibrated such that it reads 3 on the dial.

2.1 An identical weight scale is placed on the flattened nose of a rocket R1 throttled to accelerate through flat space at a magnitude of g and an identical book is placed on the scale.
2.2 A force of magnitude mg is exerted on the book
2.3 The book accelerates due to the application of the force from the rocket
2.4 The scale, being identical to the one on the table, reads 3

3.1 A rocket, R2, is identical to R1 and is also throttled such that it will accelerate at a magnitude of g when an identical scale and book are placed on its nose.
3.2 Remove any scale or book from the nose of R2, but leave the scale and book on R1, and place the two rockets nose-to-nose.
3.3 The book does not accelerate because there are equal and opposite forces of magnitude mg acting on it.
3.4 The scale reads 6.

In section 1 equal and opposite forces of magnitude mg cause the scale to read 3, but in section 3 equal and opposite forces of magnitude mg cause the scale to read 6. How can this be? Which of the numbered items above is incorrect?

 

[russ_watters]

3.3 and 3.4 are incorrect: in reality, you have no idea whatsoever what the thrust of the rocket is. The scale could read 10,000,0000 for all the information you have tells you. You're ignoring the mass of the rocket.

 

[Mechanic]

I went out into flat empty space and set the rocket throttles such that the acceleration of the books on the rocket noses was measured to be g. The mass of the book is m, so the force accelerating the book is mg, regardless of the mass of the rockets. Is this what you disagree with? I’m sure you’ve heard of the man in the rocket that is accelerating at g who cannot tell whether he is in such a rocket or on the surface of the Earth. That is the case regardless of what the mass and/or thrust of the rocket is – just as long as the acceleration is g. Are you saying that by arranging the rockets to be nose-to-nose the magnitude of the forces on the books change even though the throttles are set as described above?
Thanks

 

[russ_watters]

I went out into flat empty space and set the rocket throttles such that the acceleration of the books on the rocket noses was measured to be g.

Yes...

The mass of the book is m, so the force accelerating the book is mg, regardless of the mass of the rockets.

Correct.

Is this what you disagree with?

No. What you're missing is that the rockets are also accelerating at g, so the force applied to them is m(rocket)g. And then when you put two rockets nose to nose, you aren't just stopping the acceleration of the books, you are also stopping the acceleration of the rocket.

Consider this real-world example: The space shuttle. The space shuttle has a takeoff mass of almost exactly 2 million kg. The thrust at takeoff is:
Boosters: 12.5 million N each (2)
Main Engines: 5.4 million N total

That's an acceleration of: 1.55 g. Minus the 1g of gravity means that at takeoff the space shuttle accelerates at 0.55g.

So if you set your scale and book on top of it, it would read about 1.5N

But right before the shuttle lifts off, it is held down by explosive bolts. Alternately, you could just put a big object on top holding it down, similar to your nose-to-nose rockets. These bolts or this object pushing down on it must absorbe .55g*M(rocket), or about 11 million N of force.

 

[PJC01]

I can explain this phenomenon by considering the different scenarios presented in the content. In section 1, the book is placed on a scale on Earth, where the force of gravity (mg) is exerted on the book. The scale is calibrated to measure the weight of an object, which is the force exerted on it due to gravity. Therefore, the scale reads 3, indicating that the force exerted on the book is 3 times its mass.

In section 2, the book is placed on a scale on a rocket that is accelerating at a magnitude of g. In this scenario, the force exerted on the book is not only due to gravity, but also the acceleration of the rocket. This results in a greater force being exerted on the book, causing it to accelerate. Since the scale is identical to the one in section 1, it also reads 3.

However, in section 3, the forces acting on the book are different. The book is placed on a rocket (R2) that is identical to the one in section 2, but the scale and book are removed from the nose of R2. The scale and book are still on the nose of R1, which is now placed nose-to-nose with R2. In this scenario, the forces acting on the book are still equal and opposite, but they are now being exerted by two different sources - the gravity of Earth and the acceleration of R1. This results in a greater force being exerted on the book, causing it to accelerate more than in section 2. Therefore, the scale reads 6, indicating a greater force being exerted on the book.

None of the numbered items above are incorrect. The difference in the scale readings can be explained by the different forces acting on the book in each scenario. This highlights the importance of considering all the forces acting on an object in order to accurately measure its weight.