Exercício Dinâmica - Engenharia Mecânica (1843)

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623 
 
 
PROBLEM 13.78* 
Packages are moved from Point A on the upper floor of a warehouse to 
Point B on the lower floor, 12 ft directly below A, by means of a chute, 
the centerline of which is in the shape of a helix of vertical axis y and 
radius R = 8 ft. The cross section of the chute is to be banked in such a 
way that each package, after being released at A with no velocity, will 
slide along the centerline of the chute without ever touching its edges. 
Neglecting friction, (a) express as a function of the elevation y of a 
given Point P of the centerline the angle φ formed by the normal to the 
surface of the chute at P and the principal normal of the centerline at 
that point, (b) determine the magnitude and direction of the force 
exerted by the chute on a 20-lb package as it reaches Point B. Hint: The 
principal normal to the helix at any Point P is horizontal and directed 
toward the y axis, and the radius of curvature of the helix is 
ρ = R[1 – (h/2πR)2]. 
 
SOLUTION 
(a) At Point A: 0 0A A
A
v T
V mgh
= =
=
 
 At any Point P: 21
2PT mv= 
 2
2
1
0
2
2 ( )
P
A A P P
V Wy mgy
T V T V
mgh mv mgy
v g h y
= =
+ = +
+ = +
= −
 
 en along principal normal, horizontal and directed toward y axis 
 et tangent to centerline of the chute 
 eD along binormal 
 1 1 (12 ft)
tan tan 13.427
2 2 (8 ft)
0b
h
R
ma
β
π π
− −= = = °
=
 
 since 0ba = 
 Note: Friction is zero, 
 
2
: sin sin
: cos 0 cos
2 ( ) ( )
: 2
t t t t
b b b B
n n n
F ma mg ma a g
F ma N W N W
mv m g h y h y
F ma N W
e e e
β β
β β
Σ = = =
Σ = − = =
− −Σ = = = =