An object with a mass of 7 kg is pushed along a linear path with a kinetic friction coefficient of mu_k(x)= 4+secx. How much work would it take to move the object over x in [(-5pi)/12, (pi)/4], where x is in meters?

1 Answer
Jul 21, 2017

w_"push" = ul(7((8pi)/3vecg + (2pi)/3vecgsecx)) "J"

x in [-(5pi)/12, pi/4]

vecg > 0

And the work itself is done from the perspective of the worker.


Well, I would use a conservation of energy approach, assuming the object has no kinetic energy (i.e. it does not keep moving if we stop pushing). We just need to overcome the barrier of kinetic friction over a distance Deltax.

DeltaE = 0 = w_(vecF_k) - w_"push"

where:

  • w_"push" is the work one would have to do to push the object pi/4 + (5pi)/12 = (2pi)/3 meters forward.
  • w_(vecF_k) = vecF_kDeltax is the counteracting work done by the kinetic friction.
  • vecF_k = mu_kvecF_N is the friction force, and vecF_N is the normal force. mu_k is the coefficient of kinetic friction.

Here, we do the work from the perspective of ourselves, so w_"push" < 0 (we exert energy to do the work), but we incorporated the minus sign into the equation.

As for mu(x), we don't have to worry about domain issues, because secx is entirely continuous from start to finish in [-(5pi)/12, pi/4]. mu(x) changes as follows:

Graphed

Including the horizontal sum of the forces:

sum_i vecF_(x,i) = vecF_"push" - vecF_k = mveca_x

We realize that by not knowing the acceleration, we couldn't solve for vecF_"push" and obtain w_"push" in this manner.

We do, however, find a use in including the vertical sum of the forces:

sum_i vecF_(y,i) = vecF_N - vecF_g = vecF_N - mvecg = 0,

where our convention is that vecg > 0 (and the negative is in the subtraction sign), we have:

We now have an expression for vecF_k in terms of the mass and a constant.

color(blue)(w_"push") = vecF_kDeltax

= mu_kvecF_N Deltax

= (4 + secx)(mvecg)((2pi)/3)

= (4 + secx)("7 kg" xx g" m/s"^2)((2pi)/3 "m")

= color(blue)(7((8pi)/3g + (2pi)/3gsecx)) color(blue)("J")

The work is seen to be a function of the position (but only in [-(5pi)/12, pi/4] for its domain).

The initial push should then be hardest, the middle of the distance should be easiest, and then near the end of the distance should be slightly harder.