12. We denote the magnitude of 110N force exerted by the worker on the crate as F. The magnitude of the
static frictional force can vary between zero and fs, max =µsN.
(a) In this case, application of Newton’s second law in the vertical direction yields N=mg.Thus,
fs, max =µsN=µsmg
=(0.37)(35 kg) 9.8m/s2= 126.9N
which is greater than F. The block, which is initially at rest, stays at rest. This implies, by applying
Newton’s second law to the horizontal direction, that the magnitude of the frictional force exerted
on the crate is fs=F= 110 N.
(b) As calculated in part (a), fs, max =1.3×102N.
(c) As remarked above, the crate does not move (since F<f
s, max).
(d) Denoting the upward force exerted by the second worker as F2, then application of Newton’s
second law in the vertical direction yields N=mg −F2. Therefore, in this case, fs, max =µsN=
µs(mg −F2). In order to move the crate, Fmust satisfy F>f
s, max =µs(mg −F2), i.e.,
110 N >(0.37) (35 kg) 9.8m/s2−F2.
The minimum value of F2that satisfies this inequality is a value slightly bigger than 45.7N,sowe
express our answer as F2,min =46N.
(e) In this final case, moving the crate requires a greater horizontal push from the worker than static
friction (as computed in part (a)) can resist. Thus, Newton’s law in the horizontal direction leads
to
F+F2>f
s, max
110 N + F2>126.9N
which leads (after appropriate rounding) to F2,min =17N.