# Nature of Work done by friction

I have always been told that work done by friction can, at most, be zero, but never positive. But consider two blocks placed one on top of the other, such that their surfaces in contact are rough. If we give the block on the top a certain horizontal velocity, then in crude words, we can say that friction will try to slow down the block on the top and speed up the other one, thus opposing relative motion. Then in this case, wouldn't the work done by the friction on the block at the bottom, be positive? Please correct me if I have gone wrong.

• It depends on nature of friction if it is static in nature then work done by the friction when it is acting as a internal force is zero if it is kinetic then the work done by the friction on the whole system is non zero and always negative – Prateek Mourya Dec 10 '20 at 9:47
• You're correct. An important point, unfortunately sometimes neglected in teaching mechanics, is that work is an observer- or frame-dependent quantity, just like velocity. You can always choose a frame such that the work done by a force on a small body is positive, negative, or zero. The only quantity that is observer-independent is the total work done by all forces applied on the body, minus the change of the body's kinetic energy. Different observers will disagree on the work contributions of the various forces and on the change in kinetic energy, but they will all agree on this sum. – pglpm Dec 10 '20 at 11:53
• Isn't this "sum" zero? – nasu Dec 10 '20 at 13:39

You are completely correct, both in your description of friction and in your doubt regarding the given statement on signs.

## Regarding signs (this is purely mathematical)

The key point to be aware of is: signs have no physical meaning in themselves. They are a mathematical human invention. Signs are our human-made method for indicating direction relative to something else - that is all.

A practical way to use signs is to invent or arbitrarily choose a direction which we call positive. Any magnitude along this direction can then be considered positive and is given the sign $$+$$; any magnitude opposite to this direction can be considered negative and is given the sign $$-$$. We might need several of such reference directions (one per dimension), and together we call them a coordinate system.

Keep in mind that a coordinate system is arbitrarily chosen - it is a tool invented for our mathematical sake with no inherent physical meaning - so signs likewise are arbitrarily chosen. Claiming that something always is, say, negative is thus meanless unless we implicitly compare to something else.

So the statement:

work done by friction can, at most, be zero, but never positive

is not correct. You can choose coordinate systems at will and easily choose one that gives you a positive friction force. At the very least this statement requires some assumptions about the coordinate system's orientation to be correct.

## Regarding friction (this is physical)

we can say that friction will try to slow down the block on the top and speed up the other one

This is exactly correct. Friction will always due to its very nature pull in the direction that will prevent or stop sliding. So it always pulls so that it keeps the two surfaces together, even if this is a direction that causes motion. (It just must not cause relative motion between the two blocks, but it can cause motion in general.) This applies to both kinetic and static friction.

And you are correctly referring to Newton's 3rd law when you are saying that friction also will try to speed up the other block below. Because per Newton's 3rd law, any pull in one direction by one body comes with an equal but opposite pull in the other body. This other friction force will try to make the other block speed up, because that will help prevent sliding (it helps to keep the blocks together if the stationary block is brought along with the moving block).

Due to Newton's 3rd law we will always see a force pair pointing in opposite directions and never just a single force. So, if the coordinate system is kept constant for both bodies, then one friction force will pull in a positive direction and the other in a negative direction. Always. Only if we change the coordinate system between the two bodies can it be correct to have the friction force pointing in the negative direction in both cases.

• When I said that work done is positive, I meant that the dot product of the direction of friction and the displacement is positive. I wasn't talking about the direction of friction itself. Could you give some more clarity now, if you've understood my question? – Ambica Govind Dec 10 '20 at 16:39
• Your answer is great, it gave me a different perspective altogether, but it doesn't help me understand the flaw in my understanding. Help? – Ambica Govind Dec 10 '20 at 16:41

The energy conservation principle tells us that energy is never consumed or produced, just converted. And the very nature of friction is that it only converts from kinetic energy to heat, never the other way round.

If we ignore the heating part (as the sentence "work done by friction can, at most, be zero, but never positive" does), the total energy in the system decreases by friction.

The top block loses energy by slowing down, and the bottom block gains energy by starting to move. The stable state will be determined by conservation of momentum, and result in both blocks moving together with the same velocity. If you do the math, then you'll find that the top block loses more energy than the bottom one gains, and that loss of kinetic energy is what the sentence tries to express.

• Friction is a force that apposes the motion. It does negative work.
• The second block is subjected to reaction force due to friction and as such is not friction force.
• Just some terminology.
• The second force is considered a friction force as well. Meaning, per Newton's 3rd law when a friction force pulls one way in a body, then an equal friction force pulls the opposite way in the other body. – Steeven Dec 10 '20 at 12:30
• Also, friction is a force which opposes relative motion, not just motion in general. And that's the key here: the second friction force is causing the lower block to start moving in order to reduce the relative motion between the two blocks (it pulls the lower block along so that they won't be separated). – Steeven Dec 10 '20 at 12:32