# Work done by and against a force

What is the difference between work done by a force and work done against a force?How would we calculate the two and what is the mathematical relation between the two.

Doc Al
Mentor
Sounds like a bit of semantics. Can you give some examples of exactly what you mean?

Sounds like a bit of semantics. Can you give some examples of exactly what you mean?
For example, we can easily calculate the work done by the force of gravity in moving an object up or down an inclined plane by using the expressions W=FSCostheta or W=(-)mgh but how would we calculate the work done against the force or gravity? And what exactly does 'work done against the force of gravity' (or friction or any other force for that matter) mean?
Also, is there any relationship between the work done by and against a force? For example, are they equal and opposite or something?

Sounds like a bit of semantics. Can you give some examples of exactly what you mean?
Also, if work IS being done against a force (gravity, friction, etc), what exactly is doing the work?

Doc Al
Mentor
For example, we can easily calculate the work done by the force of gravity in moving an object up or down an inclined plane by using the expressions W=FSCostheta or W=(-)mgh but how would we calculate the work done against the force or gravity? And what exactly does 'work done against the force of gravity' (or friction or any other force for that matter) mean?
The terminology is a bit loose, but prevalent. Here's an example: You lift a box (at constant speed) a distance h. There are two forces acting on the box: gravity and the force exerted by you. The work done by gravity (as you well know) is -mgh; the work done by you, which you could think of as work done 'against gravity' is mgh.

But when work is involved, there is always something exerting a force on something else.

• D. Wani
The terminology is a bit loose, but prevalent. Here's an example: You lift a box (at constant speed) a distance h. There are two forces acting on the box: gravity and the force exerted by you. The work done by gravity (as you well know) is -mgh; the work done by you, which you could think of as work done 'against gravity' is mgh.

But when work is involved, there is always something exerting a force on something else.
How is work done by the person equal to mgh?

Doc Al
Mentor
How is work done by the person equal to mgh?
The force they exert = mg and the distance over which they exert it = h. Work = mgh.

If they exert a force greater than mg, then the work they do 'against gravity' would be the same, but they will be doing additional work to increase the kinetic energy of the box.

• D. Wani and jbriggs444
QuantumQuest
Gold Member
Simply put and according to Thermodynamics: if you supply energy through work, then work is taken as positive and when work is done by the system, it is taken as negative. If you have various forces on a system, the whole thing boils down to what force you're talking about. Producing or consuming? Then work has the appropriate sign.

• D. Wani
The work done by the force F1 is done against a force F2 when the forces F1 and F2 are directed in opposite sides one to other.

• D. Wani
The force they exert = mg and the distance over which they exert it = h. Work = mgh.

If they exert a force greater than mg, then the work they do 'against gravity' would be the same, but they will be doing additional work to increase the kinetic energy of the box.
Okay, so basically, the work done on a body against a force is the work required to move the body against the force? But just the work that is used in moving the body against the force, not the work that accelerates the body or causes its kinetic energy to increase. Did I get it right?

The work done by the force F1 is done against a force F2 when the forces F1 and F2 are directed in opposite sides one to other.
So when we say work is being done against a force, we are simply saying that work is being done on a force in the presence of another force which is directed in the opposite direction. Is this correct?

Simply put and according to Thermodynamics: if you supply energy through work, then work is taken as positive and when work is done by the system, it is taken as negative. If you have various forces on a system, the whole thing boils down to what force you're talking about. Producing or consuming? Then work has the appropriate sign.
Thank you

The force they exert = mg and the distance over which they exert it = h. Work = mgh.

If they exert a force greater than mg, then the work they do 'against gravity' would be the same, but they will be doing additional work to increase the kinetic energy of the box.
Also, how is the force the person exerts=mg.
Technically, the gravitational force=mg. If the force the person is applying is equal to mg, then the gravitational force and the person's force would cancel out leaving a net force of zero. So technically, the body shouldn't move?

Svein
Also, how is the force the person exerts=mg.
As long as the person just holds the object, yes.
So technically, the body shouldn't move?
In order to move the object, use a force greater then mg. When you are satisfied with its position, use a force slightly less than mg until the object is at rest.

cnh1995
Homework Helper
Gold Member
So technically, the body shouldn't move?
Zero net force doesn't only mean the body shouldn't move. It means, from Newton's first law, that the body maintains it's state of motion. In this case, since both the forces cancel each other, the person moves with a constant velocity. If the person were at rest, a slightly larger force than gravity would be needed to first set him into motion. That force can later be reduced slightly to make it equal to mg, when the person attains required velocity v. Now, since the net force is 0, the person will move with constant velocity v.

In order to move the object, use a force greater then mg. When you are satisfied with its position, use a force slightly less than mg until the object is at rest.
When we are calculating work done, we have to calculate the work done in actually moving the body. That means the force HAS to be greater than mg, but when we are calculating the work done against gravity, we take (mg). But if the person is only applying a force of (mg) the body won't move because the gravitational force is also (mg) and they will cancel each other out.
Automatically, the displacement becomes zero and using W=FSCostheta, the work done equals zero.
So how is it possible for the force applied by the person to be equal to the gravitational force?

Now, since the net force is 0, the person will move with constant velocity v.
Okay, so when the net force becomes zero, it keeps moving.
But then wouldn't we have to keep two forces in mind while calculating the work done? The initial force (>mg) which caused the body to move and then the force=mg which makes the net force=zero, thus keeping the body in motion

cnh1995
Homework Helper
Gold Member
But then wouldn't we have to keep two forces in mind while calculating the work done?
Yes. But if the initial force is infinitesimally greater than mg and is applied for an instant only, just to break the inertia and set the body into motion, it can be considered 0. Of course, the velocity attained is also very small(→0).

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Yes. But if the initial force is infinitesimally greater than mg and is applied for an instant only, just to break the inertia and set the body into motion, it can be considered 0. Of course, the velocity attained is also very small.
So, when we consider the force applied by the person to be zero, and we calculate work done, we are just calculating the work done in moving the body against gravity, not in increasing its kinetic energy or in accelerating it? Work done in moving a body against gravity and work done in increasing the k.e. or accelerating it are different. Is this correct?

cnh1995
Homework Helper
Gold Member
So, when we consider the force applied by the person to be zero, and we calculate work done, we are just calculating the work done in moving the body against gravity,
Yes. It's mg(h2-h1). Force applied by the person is not 0,it's mg. Net force on the body is 0. Initial kinetic energy is also 0 as v→0. The work done will be for moving the body against the gravity and the person will attain a potential energy of mgh. If the initial velocity is u and final velocity is v and no force is applied on the body except the gravity, lost KE of the body=gained PE i.e.
½mu2-½mv2=mgh. This means gravity is sucking the KE and converting it into PE, when the body goes upward against the gravity.

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cnh1995
Homework Helper
Gold Member
When the body is set into motion and the net force on the body becomes 0, it will move up with constant velocity v. The actual energy at a height h will be E=mgh+mv2/2. But since v→0 as the initial force was infinitesimally greater than mg and was applied just for an instant,
E=mgh+lim(v→0) mv2/2.
∴E=mgh.

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the work done on a body against a force is the work required to move the body against the force?
No. Absolutely. It does not metter were the body is moving. Sadnifficant is only:
when the forces F1 and F2 are directed in opposite sides one to other
And it means that the F1 works aganist to F2 and F2 works aganist to F1. Have you ever seen an arm-wrestling fight?

When the body is set into motion and the net force on the body becomes 0, it will move up with constant velocity v. The actual energy at a height h will be E=mgh+mv2/2. But since v→0 as the initial force was infinitesimally greater than mg and was applied just for an instant,
E=mgh+lim(v→0) mv2/2.
∴E=mgh.
We are concluding all of this assuming friction and air resistance are absent, right?

No. Absolutely. It does not metter were the body is moving. Sadnifficant is only:

And it means that the F1 works aganist to F2 and F2 works aganist to F1. Have you ever seen an arm-wrestling fight?
I got a little confused?

cnh1995
Homework Helper
Gold Member
We are concluding all of this assuming friction and air resistance are absent, right?
Yes. Everything is assumed to be ideal.