Work done= FXd. Why it is not= Fxdxd or FxFxd??

Question

Work done is defined by applied force on the body multplied by the distance it moves i.e.

W=F x d

which implies that W is direclty proportional to F and d.

My question is why this formula isnt

W=F x d x d ( d square) or

W=F x d x d x d ( d cube ) or

W= F x F x d

Please explain it to me with some daily problems at home??

I know this is a strange question but everything in physics has to have some logical reason to it.

Answer 1

Okay bear with me, there is a really cool explanation:

It actually come from a mathematical form called the "dot product" or

W = /F  *  /d

where W is work F is force and d is distance moved and the "/: is my way of calling F and d vectors (quantities described by two numbers, magnitude and direction).

Here Work is defined as the component of force that helped move an object in the direction of /d for a distace d.

Now a dot product is a scalar quanitity (a quantity described by only one number as opposed to vector quantities which are describe by 2 numbers). So W, or work,  is a scalar quantity.

We can calculate this dot product by using the mathematical identity of the dot product: /F * /d = |F| |d| cos(Theta), where Theta is the angle between the Force vector /F and the distance vector /d.

Now, in simple physics problems the force and the direction object moved is in the same direction - i.e. Theta = 0, or Cos(Theta) = 1

Thus W = /F * /d = |F| |d| cos(0) = |F| |d|

Or as you would write it, W=F x d !

So W=Fxd is a direct result of the simple case of the dot product definition of Work when both the force and the object's movement are in the same direction.

of course, the reason it's not F*d^2 or d^3 can also be seen as a matching of units. Energy is kg m^2/s^2 or kg m/s^2 * m.

Notice that W is a type of energy, Force has the units of kg m/s^2  and distance has units m. If it was F8d^2 then we wouldn't be describe a quantity called energy, but something like an energy-meter, which is probabaly a useless quanity to measure.

The reason energy quantity is so useful is because there are so many different types of energy and ways to formulate them, and of course the most important reason, energy is a conserved quanitity! This is the reason why we give energy so much attention and why we care that W = Fxd. I doubt F*d^2 or d^3 is a conserved quanitity so we don't care about those quantities as much.

Hope that clarifies matters.

Answer 2

To amplify what Climberguy said, work is defined like it is because it is useful in mechanics and thermodynamics.

To turn the question around, what would Fd^2 mean? Could you better understand physical properties and changes with this definition of work? The same goes for your other alternative definitions, they don't correspond to any behavior in the real world.

The history of the definition of work linked below may help you understand how we arrived at that definition and what it means.

Answer 3

work is what we say it is, just as volume is what we say it is and force is what we say it is. we say that work is force times distance, so that is what it is. this is how things work. its really the units that matter, because they are what make it work with everything else.

make it a good day

Answer 4

Think about it like this.  How much more work does it take you to carry a book up one flight of stairs versus two flights of stairs.  Most people would say, not even knowing physics but just from experience, that it takes twice as much work to carry it up two flights.  Physics usually matches intuition and it does here too-- to apply a force (such as lifting a book) twice as far is twice as much work.