Solução_Calculo_Stewart_6e

F.
TX.10
PROBLEMS PLUS ¤ 137
steeper the inclined plane, the less the horizontal components of the various forces affect the movement of the block, so we
would need a much larger magnitude of horizontal force to keep the block motionless. If we allow θ → 90 ◦ , corresponding
to the inclined plane being placed vertically, the value of h min is quite large; this is to be expected, as it takes a great
amount of horizontal force to keep an object from moving vertically. In fact, without friction (so θ s =0), we would have
θ → 90 ◦ ⇒ h min →∞, and it would be impossible to keep the block from slipping.
(d) Since h max is the largest value of h that keeps the block from slipping, the force of friction is keeping the block from
moving up the inclined plane; thus, F is directed down the plane. Our system of forces is similar to that in part (b), then,
except that we have F = −(μ s n) i. (Note that |F| is again maximal.) Following our procedure in parts (b) and (c), we
equate components:
−μ s n − mg sin θ + h max cos θ =0 ⇒ h max cos θ − μ s n = mg sin θ
n − mg cos θ − h max sin θ =0 ⇒ h max sin θ + mg cos θ = n
Then substituting,
h max cos θ − μ s (h max sin θ + mg cos θ) =mg sin θ
⇒
h max cos θ − h max μ s sin θ = mg sin θ + mgμ s cos θ
⇒
sin θ + μs cos θ tan θ + μs
h max = mg
= mg
cos θ − μ s sin θ 1 − μ s tan θ
tan θ +tanθs
= mg
= mg tan(θ + θ s )
1 − tan θ s tan θ
We would expect h max to increase as θ increases, with similar behavior as we established for h min ,butwithh max values
always larger than h min .Wecanseethatthisisthecaseifwegraphh max as a function of θ,asthecurveisthegraphof
h min translated 2θ s to the left, so the equation does seem reasonable. Notice that the equation predicts h max →∞as
θ → (90 ◦ − θ s ). In fact, as h max increases, the normal force increases as well. When (90 ◦ − θ s ) ≤ θ ≤ 90 ◦ ,the
horizontal force is completely counteracted by the sum of the normal and frictional forces, so no part of the horizontal
force contributes to moving the block up the plane no matter how large its magnitude.