A shopping bag is hanging straight down from your hand as you walk in horizontal floor at aconstant velocity.

Question

Does the force that your hand exerts on the bag's handle do any work?Explain

Answer 1

No, it does not.
For a force to do any work it must have a component in the direction of the displacement. This is obvious from the definition of work as the inner product of the Force vector by the Displacement vector ie F r cos (angle) where (angle) is the angle between the direction of the force and of the displacement. When angle = 90 degrees (as is your case) then cos(angle) = 0 and the work is zero.

Answer 2

If you are not swinging your arm/bag while walking at the constant velocity, the only force (of note) acting on the bag is that of gravity (W). The only counterforce, to keep the bag in your hand, is the equal and opposite force (F) exerted by your fingers.

Thus, as you move along unnatrually, because you're not swinging your arm as you walk, we have this f = ma = (W - F); where m is the mass of your bag and a is its acceleration (up/down). f is the net force you were asking about.

If W = F, what is the force f on that bag? That's f = ma = (W - F) = 0. So the net force f = 0 in the up/down direction. If f <> 0, that handbag would be accelerating in either the up or down direction depending on which is bigger W or F. But it isn't accelerating, it's snug in your hand.

Now as to work. The work function WE = fS; where f is the same one in f = ma and S is the distance the handbag is moved in the direction of and under the push of f. f is acting up/down; so S is up/down only, which way are you walking? You're walking horizontally; so S = 0 and f = 0. That's about as zero as one can get for work WE in the vertical direction.

What about work in the horizontal direction. Given the previous assumptions, there is no force (F = 0) acting in the horizontal direction D. So we = FD = 0*D = 0 and walking does no work on your bag. That is, because there is zero force in the direction you are walking, there is no work being done on the bag there either.

[Note, if you were not walking at a constant velocity, F <> 0 = ma, where a = dv/dt a change in velocity called acceleration. In which case we = FD <> 0 and there would be work horizontally. In fact, if the bag is allowed to swing upward as you accelerate, we can show there would be work vertically as well.]

Bottom line, under the conditions you describe, you do no work on the bag...period. There is no work in the vertical direction because there is no net force and there is no vertical distance covered. There is no work in the horizontal direction because there are no horizontal forces on the bag.

[Another note: There was work done on the bag...the work of raising the bag to the height h of your hand. In which case WE = Wh = mgh = PE; where the work force is the weight of the bag W and the distance moved is h, the height of the hand. Also, the bag now has potential energy PE = mgh as the result of that earlier work. So should you drop the bag, that potential energy will become kinetic energy, KE.]