Because the block is on the verge of moving, the static frictional force must be at its maximum possible value, that is, `f_(s)=f_(s.max)`. (2) Because the block is on the verge of moving up the plane, the frictional force must be down the plane (to oppose the pending motion). (3) From Sample Problem 5.05, we know that the component of the gravitational force down the plane is `mgsintheta` and the component perpendicular to (and inward from the plane is `mgcostheta` (Fig. 6-11b).
Calculations: Figure 6-11c is a free-body diagram for the block, showing the force `vecF` applied by the ropes, the static frictional force `f_(s)`, and the two components of the gravitational force. We can write Newton.s second law `(vecF_("net") =mveca)` for forces along the x axis as
`F-mgsintheta-f_(s)=m(0)`
Because the block is on the verge of sliding and the frictional force is at the maximum possible value `f_(s.max)` we use Eq. 6-1 to replace `f_(s)` with `mu_(s)F_(N)`:
`f_(s)=f_(s.max)`
`=mu_(s)F_(N)`
From Fig. 6-11c, we see that along the y axis Newton.s second law becomes
`F_(N)-mgcostheta=m(0)`.
Solving Eq. 6-21 for `F_(N)` and substituting the result into Eq. 6-20, we have
`f_(s)=mu_(s)mgcostheta`.
Substituting this expression into Eq.6-19 and solving for F lead to
`F=mu_(s)mgcostheta+mgsintheta`
Substituting `m=200kg,theta=52^(@),andmu_(s)=0.40`, we find the stone block on the verge of moving is `2.027xx10^(4)N`. Dividing this by the assumed pulling force of 686 N from each man, we find that the required number of men is
`N=(2.027xx10^(4)N)/(686N)=29.5~~30` men.
Comment: Once the stone block began to move, the friction was kinetic friction and the coefficient was about 0.20. You can show that the required number of men was then 26 or 27. Thus, the huge stone blocks of the Great Pyramid could be pulled up into position by reasonably small teams of men.
Figure 6-11 (a) A stone block on the verge of being pulled up the side of the Great Pyramid. (b) The components of the gravitational force. (c) A free-body diagram for the block.
