# Work done against a force

Consider a body of $1\ \rm kg$ mass at rest at ground. Now as we know, it can be lifted by a force equal to its weight. So we have to apply a little bit more force then its weight. let $g= 10\ \rm ms^{-2}$, its weight is $10\ \rm N$. So I apply a force of $20\ \rm N$ or $10\ \rm N$ net force. As we know all the work I shall do will be converted to P.E of the mass. In case of friction my work done is wasted as heat.But what in the case if we consider a frictionless world for a while and there is no gravity now if we consider a mass moving with constant velocity in space with $10\ \rm N$ force applied on it and an opposing force of $-10\ \rm N$ acting on it where the work done by $10\ \rm N$ force against $-10\ \rm N$ will go?

• It will go into the object, which supplies the 10 N. That pulling force must come from somewhere - maybe a rocket (so the energy is spent to burn and accelerate fuel) or a gravitational field (so the energy is spent "moving" the planet / stored as potential energy) or alike – Steeven Aug 7 '18 at 13:12
• Just to be clear, are you saying there are no forces (gravity, friction, etc.) acting on the mass except for 2 equal and opposite forces and the velocity is constant? – Bob D Aug 7 '18 at 13:28
• Of curse! I an talking about the same thing – AHTSHAM KHALID Aug 25 '18 at 13:28
• Please, separate your question in paragraphs, and use commas and full stops. If you make reading easy, you'll surely get better and more useful answers. Try to state your question in a calm and organised way. – FGSUZ Sep 1 '18 at 22:52

The formula for work is:

The work $W$ done by a constant force of magnitude $F$on a point that moves a displacement $s$ in a straight line in the direction of the force is the product $W = F \cdot s$.

When we are talking about the world without friction, and the body (point) is moving straight with constant velocity we can say in accordance with the Newton's first law that:

In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.

In your case where two equal forces $\vec{F_1} = -\vec{F_2}$ act on a point in opposite directions, in accordance with the superposition principle: $\vec{F_1} +\vec{F_2} = 0$.

Then based on $\vec{F} = m\vec{a}$, follows $\vec{a} = 0$ and velocity is a constant, moreover by inertial frame introduction above we can set $s = 0$, in inertial frame, where point is at a rest.

If we talk about $F_1$, and we want to calculate the work $W_1$ done by $F_1$, using the formula above we get $W_1 = F_1 \cdot 0 = 0$.

• Just because net force is $0$ does not mean displacement is $0$. – BioPhysicist Sep 1 '18 at 22:08
• Corrected an answer about inertial frame – Artem Sep 1 '18 at 22:12
• "if we consider a mass moving with constant velocity". The OP is considering a scenario where we are in an inertial frame and we observe the object moving at constant velocity with two equal but opposite forces acting on it. I can see what you are doing here now, but you should explicitly tell the OP in your answer why this new scenario you have created still answers the original question. – BioPhysicist Sep 1 '18 at 22:21

In the case of an object moving at a constant velocity with two equal but opposite forces acting on the object, the net work done on the object is $W=\int \vec F_{net} \cdot d\vec s= \int \left (\vec F_1-\vec F_2\right ) \cdot d\vec s=0$. One force inputs energy and the other removes energy.

Without knowing more about the forces in question we cannot say where the energy is coming from or going to. For example if we are dealing with two conservative forces then the potential energy from one force is being converted into potential energy of the other force. If we allow friction back into our world and one of the forces is non-conservative (like some dissipative force), then the energy will be lost to heat.

All we can say is that the object will not be changing its kinetic energy.

• OP indicated that its a "frictionless world" so all forces are conservative. – Artem Sep 1 '18 at 22:47
• @Artem Thanks. I have addressed this in my answer. – BioPhysicist Sep 1 '18 at 22:51

Good question!! And the answer is simple but interesting!

In your case, the work done by the two forces will be stored in that body. But the interesting thing is that: what will be the amount of work done by these two forces on the body?

Before we proceed, you must understand these five things:

(1) Work will be done by a force only when some kind of displacement is caused by it. If it doesn't cause any displacement, it doesn't do any work.

(2) The displacement caused can be outside eg. displacement of the body from point A to B. Or it can be inside of the body eg. compression of the body by 1 mm.

(3) The work (or we can say energy) is generally stored in a body in three forms: Potential energy, Kinetic energy, and Internal energy.

(4) The PE and KE is measured with respect to the surrounding because height or speed is always with respect to surrounding. The internal energy is not measured with respect to the surrounding but with respect to it's initial phase/ energy level.

(5) Further, the internal energy can be stored by a body in two forms: (a) reversible or elastic or temporary eg. potential energy stored in a compressed spring. (b) irreversible or inelastic or permanent eg. slight heating of the spring due to continuous compression.

Now let's analyze your case. The two equal & opposite forces are being applied on the body from opposite sides. These two forces will try to compress the body. Now it depends on (1) the material of the body (2) the magnitude of the applied forces and (3) the duration for which the forces are being applied. These three things will decide that how much compression is caused in the body i.e. whether it will be in mm or micrometre or nanometre... and whether it will be elastic or inelastic... And all that energy will keep on getting stored inside that body.

This continuous force/pressure over very very long period of time can even cause metamorphic changes inside that body. For ex. formation of coal from wood, formation of metamorphic rocks, formation of stars and planets etc..