Problem 9.12 The mass of the box on the left is 30 kg,
and the mass of the box on the right is 40 kg. The coefficient of static friction between each box and the inclined
surface is &s D 0.2. Determine the minimum angle ˛ for
with the boxes will remain stationary.
α
Solution: If the boxes slip when ˛ is decreased, they will slip
toward the right. Assume that slip toward the right impends, the free
body diagrams are as shown.
y
y
T
α
(30)(9.81)
The equilibrium equations are
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30°
Fx D T % 0.2 NA % ,30+,9.81+ sin ˛ D 0,
(1)
Fy D NA % ,30+,9.81+ cos ˛ D 0,
(2)
Fx D %T % 0.2 NB C ,40+,9.81+ sin 30° D 0,
(3)
Fy D NB % ,40+,9.81+ cos 30° D 0,
(4)
0.2 NA
NA
x
T
30°
(40)(9.81)
0.2 NB
NB
x
Summing Equations (1) and (3), we obtain %0.2 NA % 0.2 NB %
,30+,9.81+ sin ˛ C ,40+,9.81+ sin 30° D 0. Solving Equation (2) for
NA and Equation (4) for NB and substituting the results into
°
Equation (5) gives
" 15 sin ˛ C 3 cos ˛ D 10 % 4 cos 30 . (6) Using the
identity cos ˛ D 1 % sin2 ˛ and solving Equation (6) for sin ˛, we
obtain sin ˛ D 0.242, so ˛ D 14.0°
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