Why do we multiply force and displacement in physics?

[Atomic_Sheep]

Wikipedia says:

The work done by a constant force of magnitude F on a point that moves a displacement (not distance) s in the direction of the force is the product

{\displaystyle W=Fs}[PLAIN]https://wikimedia.org/api/rest_v1/media/math/render/svg/b020230128c115d1b9e9cfbe6636985e98bbaf69.
The question is this...

What's the intuition behind multiplying force by displacement? You learn multiplication tables etc but then you start getting into multiplying abstract things like force and displacement? Why does multiplying these things work?

 

[QuantumQuest]

Wikipedia says:

The work done by a constant force of magnitude F on a point that moves a displacement (not distance) s in the direction of the force is the product

{\displaystyle W=Fs}[PLAIN]https://wikimedia.org/api/rest_v1/media/math/render/svg/b020230128c115d1b9e9cfbe6636985e98bbaf69.
The question is this...

What's the intuition behind multiplying force by displacement? You learn multiplication tables etc but then you start getting into multiplying abstract things like force and displacement? Why does multiplying these things work?

In Physics, there are scalar and vector physical quantities. Beginning from fundamental quantities (time, distance, mass etc.), we derive more complex ones.

To have an intuition about work, you must ask yourself what does this quantity represent. It expresses transfer of energy or transformation of energy from one form to another. So, it has units of energy (Joule in S.I). The equation of work $W = F\cdot s\cdot cosa$ or $W =F\cdot s$ for the case that force has the direction of motion, expresses such energy.

 

[PeroK]

Wikipedia says:

The work done by a constant force of magnitude F on a point that moves a displacement (not distance) s in the direction of the force is the product

{\displaystyle W=Fs}[PLAIN]https://wikimedia.org/api/rest_v1/media/math/render/svg/b020230128c115d1b9e9cfbe6636985e98bbaf69.
The question is this...

What's the intuition behind multiplying force by displacement? You learn multiplication tables etc but then you start getting into multiplying abstract things like force and displacement? Why does multiplying these things work?

In what way are force and displacement abstract? Surely these are very real things from the physical world?

 

[Chestermiller]

Is your question something like "Why is mathematics able to describe physical phenomena?"

 

[David Lewis]

Force and displacement follow the rules of multiplication when calculating work. For example, if you double the force, you double the work. If you double the force, and triple the displacement, you have 6 times as much work, and so on.

This was an unexpected application of multiplication tables. Multiplication's original purpose was a shortcut to adding like things. (Whenever you add physical quantities they must all be the same quantity.) When you multiply them, however, it turns out they don't have to be.

 

[Atomic_Sheep]

In Physics, there are scalar and vector physical quantities. Beginning from fundamental quantities (time, distance, mass etc.), we derive more complex ones.

Reading about this... very interesting. Exactly the info that I was looking for.

In what way are force and displacement abstract? Surely these are very real things from the physical world?

If you're counting apples... 2 apples per bag x 1 bag... = 2 apples or 1 bag. How can you multiply a bag by an apple? You've got abstract things like bags apples force displacement. Then if you multiply 2 apples by 1 bag you get 2 apples or 1 bag, so 1x2 = 2 and 1. Perhaps very broken logic but just my interpretation of it.

Is your question something like "Why is mathematics able to describe physical phenomena?"

I haven't thought about it that way.

Force and displacement follow the rules of multiplication when calculating work. For example, if you double the force, you double the work. If you double the force, and triple the displacement, you have 6 times as much work, and so on.

This was an unexpected application of multiplication tables. Multiplication's original purpose was a shortcut to adding like things. (Whenever you add physical quantities they must all be the same quantity.) When you multiply them, however, it turns out they don't have to be.

Yep, this is partly what is bugging me, why do the things you multiply not have to be the same. I guess I just need to accept this fact and move on.

 

[PeroK]

Yep, this is partly what is bugging me, why do the things you multiply not have to be the same. I guess I just need to accept this fact and move on.

What is apple x apple? Or, Force x Force?

You say you have to accept this fact as though it was something really odd. But, to take division, if you have £50, you can divide that between 5 people and they get £10 each. Why is that something abstract that you just have to accept?

 

[David Lewis]

When you multiply, the physical quantities also multiply (not just the numbers). For example, length x length = area. The product is a new physical quantity.

 

[gmax137]

... if you double the force, you double the work. If you double the force, and triple the displacement, you have 6 times as much work, and so on...

This is the important point. If you don't feel it intuitively, think about carrying a sack of concrete up the stairs. If you have to go up two floors, that's "twice as hard" as going up one floor. Or, carrying two sacks is "twice as hard" as carrying one sack. The physics definitions of work idealize the situation and remove the effects of human anatomy, but the idea is the same. This means WORK is proportional to both force and distance. Mathematically, you multiply them together. This produces "funny units" like Newton-meters or foot-pounds. What do these mean? They mean two flights of stairs is twice the work of one flight. Or, two flights with a 50 pound sack is the same as one flight with a 100 pound sack.

 

[kuruman]

Yep, this is partly what is bugging me, why do the things you multiply not have to be the same. I guess I just need to accept this fact and move on.

Multiplying different quantities together to get a new quantity that describes a new idea is very common in physics but is also encountered in the "real" world. When one says that it takes 12 person-hours to build a wall, the meaning is clear: the wall can be built by 1 person in 12 hours or by 3 persons in 4 hours, etc.. Clearly, a person-hour is a separate unit from a unit of humanity (person) and from a unit of time (hour).

 

[Atomic_Sheep]

What is apple x apple? Or, Force x Force?

Well if you have 5 apples and another 5 apples and you multiply them, then you have 10 apples + 15 magically created apples :), I'm not trolling I promise.

So what do you get when you multiply apples by apples or force by force... good question... no idea. I guess what kuruman said...

Multiplying different quantities together to get a new quantity that describes a new idea is very common in physics but is also encountered in the "real" world.

We should get a new quantity, although in this case we're multiplying like with like so perhaps not. hmm...

if you have a force of 1N and another one 1N, if you add them you get 2N but if you multiply you get 1N
if you have a force of 2N and another one of 2N, if you add them, you get 4N and if you multiply them you get 4N
if you have a force of 3N and another one of 3N, if you add them, you get 6N and if you multiply them you get 9N

So to me it seems like these operations can't be compared directly. If you have these forces as vectors, the addition makes sense, the vectors simply add up and the forces become larger. However if you multiply, you get mixed results... at 1N, you have less force than if you added them, at 2N, they are equal to addition, at 3N and above, multiplication appears to create magical forces out of nowhere and in the case of 1N you actually get no extra force and appear to lose 1N. Same as the apple example above.

 

[A.T.]

We should get a new quantity, although in this case we're multiplying like with like so perhaps not.

If you multiply meters with meters to compute an area, you get square meters, which is a different unit.

What might be confusing in the discrete case: If you multiply apples with apples to compute the number of apples arranged in a rectangular grid, you still get apples. That's because what you actually do here is to multiply apples/row with rows.

 

[gmax137]

...We should get a new quantity, although in this case we're multiplying like with like so perhaps not. hmm...

if you have a force of 1N and another one 1N, if you add them you get 2N but if you multiply you get 1N
if you have a force of 2N and another one of 2N, if you add them, you get 4N and if you multiply them you get 4N
if you have a force of 3N and another one of 3N, if you add them, you get 6N and if you multiply them you get 9N

No no no!

If you have a force of 1N and another one 1N, if you multiply you get 1*1*N*N or $1N^2$

The units get multiplied just like the values.

Now, $Newton^2$ doesn't make a lot of sense, and you probably wouldn't see multiplication of force like that. But $meter^2$ (area) is common, as is Newton*meter(work or energy) and any number of other compound unit parameters (man*hours, $ft/sec$, or even $Btu/hr~ft^2$)