# Force and acceleration (not actually homework)

• ViolentCorpse
In summary, the conversation discusses a conceptual problem regarding the weight and acceleration of a stack of books. Two different approaches to finding the total weight are discussed - one simply adds the masses together, while the other adds the extra acceleration caused by the top book pushing down on the bottom book. The conversation also touches on the interpretation of gravity and normal force, and how they relate to the acceleration of objects. Ultimately, there is no contradiction in the two approaches, as they lead to the same answer.
ViolentCorpse

## Homework Statement

This is more of a conceptual problem than a mathematical one (Note that this is purely for conceptual purposes of my own and probably doesn't really hold any practical value). I was just curious to know that if we imagine a book of mass m1 resting on any flat surface on Earth and then put on top of it another book of mass m2, then the greater weight of the stack obviously seems a direct result of the additional mass (the book) added on top of the bottom book. But then a thought occurred to me: wouldn't the book on top pressing down on the book on bottom increase the downward acceleration of the bottom book? I turned to mathematics for the answer and I applied two slightly different approaches. Though, the two approaches give the same answer for the total weight of the stack, the values for acceleration that I got don't quite add up.

The two approaches I applied are:

1)Simply adding the two masses together and multiplying by g i.e
F=(m1+m2)*g

2)Finding the extra acceleration a that is caused by the top book pushing down on the bottom book, then adding the resulting value to g and multiplying by the mass of the bottom book only i.e
F=m1*(g+a)

It seems the problem can be interpreted in two ways that appear, at least on the face of it, equivalent. The first approaches imagines a single system of two books. The second approach reduces the physical book to a downward force, so we're effectively dealing with a single book.

But there must a single correct physical interpretation to this. Is the bottom book being accelerated by 9.8m/s^2 or something more? That's my question.

P.S: I am aware that this is an absolutely pointless exercise, since the net accelerations, whatever the component accelerations, are always going to cancel out in this case no matter what approach we take.

Thank you.

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ViolentCorpse said:
But then a thought occurred to me: wouldn't the book on top pressing down on the book on bottom increase the downward acceleration of the bottom book?
If the books are resting on a surface their acceleration is zero.

But there must a single correct physical interpretation to this. Is the bottom book being accelerated by 9.8m/s^2 or something more? That's my question.
Think of 9.8 m/s^2 as a measure of the strength of Earth's gravitational field. It's the acceleration of a falling body when the only force acting is gravity, which is not the case here. But the weight of an object is mg, regardless of its acceleration.

1 person
Doc Al said:
If the books are resting on a surface their acceleration is zero.

I meant that in the sense that we're only looking at the component acceleration only. The net acceleration would of course be zero (for example, 13m/s^2 of acceleration of a single object being canceled by an equal and opposite force). But I wanted to know what these equal and opposite acceleration values are, if looked in isolation.

Doc Al said:
Think of 9.8 m/s^2 as a measure of the strength of Earth's gravitational field. It's the acceleration of a falling body when the only force acting is gravity, which is not the case here. But the weight of an object is mg, regardless of its acceleration.
I agree about the weight, but it's specifically the acceleration I'm concerned about.

ViolentCorpse said:
wouldn't the book on top pressing down on the book on bottom increase the downward acceleration of the bottom book?

We will drop the two books from a building(air resistance is negligible)
The acceleration of two books will be constant in this case.Isn't it?
Even if we add the masses,the acceleration will still be constant.

1 person
ViolentCorpse said:
I meant that in the sense that we're only looking at the component acceleration only. The net acceleration would of course be zero (for example, 13m/s^2 of acceleration of a single object being canceled by an equal and opposite force). But I wanted to know what these equal and opposite acceleration values are, if looked in isolation.
Do not think in terms of 'component acceleration'. Think in terms of forces. Then apply Newton's 2nd law to find the resulting acceleration.

To think that gravity creates a 9.8 m/s^2 acceleration downward and the normal force creates a 9.8 m/s^2 acceleration upward (for a net of zero) is a confusing an inaccurate way of viewing things.

1 person
ViolentCorpse said:

## Homework Statement

1)Simply adding the two masses together and multiplying by g i.e
F=(m1+m2)*g

2)Finding the extra acceleration a that is caused by the top book pushing down on the bottom book, then adding the resulting value to g and multiplying by the mass of the bottom book only i.e
F=m1*(g+a)

Thank you.

But if you REALLY want to go this way, there is no contradiction.
That "a" in the formula is the acceleration of m1 under the weight of m2. So it will me (m2g)/m1
Plug in and you get the same as in (1).

1 person
Doc Al said:
Do not think in terms of 'component acceleration'. Think in terms of forces. Then apply Newton's 2nd law to find the resulting acceleration.

To think that gravity creates a 9.8 m/s^2 acceleration downward and the normal force creates a 9.8 m/s^2 acceleration upward (for a net of zero) is a confusing an inaccurate way of viewing things.
I see. That's a very helpful instruction.

Thanks a lot guys! :)

## 1. What is the relationship between force and acceleration?

The relationship between force and acceleration is described by Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force acting on the object and inversely proportional to its mass. In simpler terms, the more force applied to an object, the greater its acceleration will be.

## 2. How is force and acceleration calculated?

Force can be calculated using the equation F = ma, where F is force in Newtons, m is mass in kilograms, and a is acceleration in meters per second squared. Acceleration can also be calculated using the equation a = F/m, which rearranges the original equation.

## 3. What is the difference between mass and weight?

Mass is a measure of the amount of matter in an object, while weight is a measure of the force of gravity acting on an object. Mass is typically measured in kilograms, while weight is measured in Newtons. On Earth, an object's weight is directly proportional to its mass, but on other planets, the force of gravity may be different and therefore, an object's weight would also be different.

## 4. How does friction affect force and acceleration?

Friction is a force that opposes motion and is caused by the interaction of surfaces. When friction is present, it will reduce the net force acting on an object and therefore, decrease its acceleration. This is because some of the force is being used to overcome the force of friction instead of accelerating the object.

## 5. Can an object have a constant acceleration and changing force?

Yes, an object can have a constant acceleration and changing force if the direction of the force is changing. This can occur when an object is moving in a circular path, as the direction of the force is constantly changing, but the magnitude of the force may remain the same. This is described by Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration, regardless of the direction of the force.

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