Why Do We Multiply Force and Displacement? Exploring Work and Energy

[rishch]

These are some of my question related to work and energy-

1)W is proportional to Force
W is proportional to displacement
Therefore, W=Fs
My question is why do we have to multiply F and s.Why can't we add them.Why can't the formula be W=F+s
2)In space if you were to push an object it would keep on going.Displacement could be infinite.Therefore, the work you did would be infinite and therefore your energy as well.How can you have infinite energy?
3)Suppose you have 2 balls of the same mass.One is already in motion while the other is at rest.You apply a force F on Both the balls for the same time period.If you calculate the work done in both the cases the work done in the first case would be greater because the distance traveled in the time in which the force was applied would be greater as the ball was aleady in motion.Shouldn't the work be the same?If you were to push 2 identical balls with the same force how would you use greater energy in one case.Isn't the extra displacement because the ball was already in motion due to a previous force?
4)The derivation for the formula for potential energy in my book goes like this-
The potential energy of an object above the ground is equal to the work done in raising it from the ground to that point against the force of gravity.So the displacement would be its height from the ground h.The minimum force required to raise the object would be mg.Therefore P.E=mgh.How can the minimum force required to lift the object be mg.The weight of the object would also be mg amd the two forces should cancel each other out.Yet they say that the object is raised with no force.How?
5)Suppose you raise an object to a certain height.Its P.E would be mgh.But now if you were to dig a pit directly under the ball the height would increase and therefore, so would the P.E.But no work was done on the ball and still there was an increase in energy.My book says change in energy of a body=work done on that body.But in this case no work was done and still there was an increase in energy.How?
6)Imagine two people standing opposite to each other.hey both begin to push each other with same force.After some time they would both feel tired but as there was no displacement there was no work done.So where did there energy go?Doesn't this violate the law of conservation of energy?
7)In my book when introducing the concept of power they say-"Imagine two people A and B of same mass.They both cover a distance s but A does it in 15 seconds while B does it in 20 seconds.Work done is same but A does it in a shorter time.Who would you say is more powerful?"
My question is -How can the work done be the same.If A reaches faster then he must have used greater force so as to accelerate faster.How is Work the same?
8)My book says P.E does not matter on the path taken to reach the point.But suppose you were to raise an object to a height h but not straight up through say a staircase.Wouldn't the work done on the object be more than if you were to raise it straight up as the diagonal path would be longer than the straight path so displacement would be more?
9)If you place your hand in hot water your hand warms up.There is transfer of energy.My book says change in energy of a body=work done on that body.So what's the work done?

 

[JHamm]

1) You COULD define it this way, but it wouldn't help you much, I think what you're wanting to ask is WHY it is useful to consider that definition. First that definition of work only applies when the force is a constant, the actual definition is an integral of the force through the displacement [itex]W = \int_a^bf(x)dx[/itex]. In this way it is really the sum of all the tiny pushes you make.

2, 3) You seem to have muddled up the definition of work a little, the displacement you multiply by isn't the displacement of the object after you push it (if it were you'd do more work pushing a ball down a hill than up since it would go further) but rather the distance over which the force is being applied, so for the work to be infinite you would need to keep applying your force over an infinite distance.

4) The idea is somewhat confusing that way, the real way to define potential energy is the work that had to be done moving the object to that height, this happens to be (by a theorem you'll learn later) equal to the work that the Earth would do bringing the object down by that height. Since the force bringing it down is mg and the distance is h you have your mgh.

5) Potential energy is a relative concept, you can define your zero anywhere as long as you keep it constant through the problem you work on, so if you take ground level to be at zero potential energy then if you raised a ball and dropped it into a hole it's new potential energy in the hole would be negative (you'll find out that it is only the changes in PE that matter, not the value itself so there is no real problems with negatives)

6) I have a bit of a problem when examples involving people pushing are used because it does tend to be very confusing for this very reason, you feel tired because there actually IS being work done in your muscles as they continually contract and relax to maintain a constant push, but in a physics class this is usually ignored (for better or for worse).

7) The work is the same because the person who did it in less time used less force at the end of the run as he was slowing down, think of how it feels to come out of a sprint, you might cover 20 or so meters and feel like you're barely moving but if you cover that distance when jogging it feels noticeably harder.

 

[Doc Al]

These are some of my question related to work and energy-

1)W is proportional to Force
W is proportional to displacement
Therefore, W=Fs
My question is why do we have to multiply F and s.Why can't we add them.Why can't the formula be W=F+s

For one thing, with such an equation work would no longer be proportional to either F or s. In general: it makes no physical sense to add quantities of different kinds, that don't even have the same units.

2)In space if you were to push an object it would keep on going.Displacement could be infinite.Therefore, the work you did would be infinite and therefore your energy as well.How can you have infinite energy?

Let's bring your example down to earth. Say you have a box on a frictionless floor. Give it a push and it will keep moving forever without any force or work needed to continue the motion. Now if you want to keep pushing the box, so it goes faster and faster, realize that that becomes harder and harder. It will require infinite energy to do that--which is another way of saying you won't be able to do it.

3)Suppose you have 2 balls of the same mass.One is already in motion while the other is at rest.You apply a force F on Both the balls for the same time period.If you calculate the work done in both the cases the work done in the first case would be greater because the distance traveled in the time in which the force was applied would be greater as the ball was aleady in motion.Shouldn't the work be the same?If you were to push 2 identical balls with the same force how would you use greater energy in one case.Isn't the extra displacement because the ball was already in motion due to a previous force?

Go back to the example with the box. Realize that it's harder to push a moving box. For the same time, you end up having to push it a longer distance which means more work is done.

4)The derivation for the formula for potential energy in my book goes like this-
The potential energy of an object above the ground is equal to the work done in raising it from the ground to that point against the force of gravity.So the displacement would be its height from the ground h.The minimum force required to raise the object would be mg.Therefore P.E=mgh.How can the minimum force required to lift the object be mg.The weight of the object would also be mg amd the two forces should cancel each other out.Yet they say that the object is raised with no force.How?

You're right. In order to start lifting the object you have to exert slightly more force than the weight of the object. But to just barely lift it with the least amount of added speed, we think of just using a force equal to its weight.

5)Suppose you raise an object to a certain height.Its P.E would be mgh.But now if you were to dig a pit directly under the ball the height would increase and therefore, so would the P.E.But no work was done on the ball and still there was an increase in energy.My book says change in energy of a body=work done on that body.But in this case no work was done and still there was an increase in energy.How?

There was no change in PE. The PE with respect to a point underneath the Earth is the same whether or not you actually dig the hole. Imagine a book on a table. If you measure the gravitational PE from the table top, it's zero. But measure it from the floor and it's not zero, but mgh. The point is that only changes in PE matter. And until you move the object, no change in PE takes place.

6)Imagine two people standing opposite to each other.hey both begin to push each other with same force.After some time they would both feel tired but as there was no displacement there was no work done.So where did there energy go?Doesn't this violate the law of conservation of energy?

Just because no mechanical work was done, does not mean that energy wasn't expended. Realize that people are biological systems. Another example: You hold up a weight as long as you can. The weight doesn't move, so you do no work on the weight. Yet it requires energy for you to maintain the tension in your arms. But that's biology, not simple physics. (Work is being done within your muscles, as the fibers contract and release--yet you do no work on the weight.) You could have just put the weight on a table--the table would support the weight without needing any energy input.

7)In my book when introducing the concept of power they say-"Imagine two people A and B of same mass.They both cover a distance s but A does it in 15 seconds while B does it in 20 seconds.Work done is same but A does it in a shorter time.Who would you say is more powerful?"
My question is -How can the work done be the same.If A reaches faster then he must have used greater force so as to accelerate faster.How is Work the same?

Work is force X displacement. It doesn't matter how fast you move, just how far.

8)My book says P.E does not matter on the path taken to reach the point.But suppose you were to raise an object to a height h but not straight up through say a staircase.Wouldn't the work done on the object be more than if you were to raise it straight up as the diagonal path would be longer than the straight path so displacement would be more?

Realize that when using the formula W = F*s, you need the component of the force in the direction of the displacement. A more complete formula might be: W = F*s*cosθ, where θ is the angle between the force and the displacement.

9)If you place your hand in hot water your hand warms up.There is transfer of energy.My book says change in energy of a body=work done on that body.So what's the work done?

The change in mechanical energy equals the work done. But there are other forms of energy transfer, such as heat flow, where no work is done.

Good questions!