PROBLEM 13.62
An elastic cable is to be designed for bungee jumping from a
tower 130 ft high. The specifications call for the cable to be 85
ft long when unstretched, and to stretch to a total length of 100
ft when a 600–lb weight is attached to it and dropped from the
tower. Determine (a) the required spring constant k of the
cable, (b) how close to the ground a 186–lb man will come if
he uses this cable to jump from the tower.
PROBLEM 13.62 (Continued)
d=
distance from the ground
112 2
TVTV+=+
2
2
0 (186)(130 ) 0 (266.67)(45 )
266.7 23815 515827 0
dd
dd
+ −=+ −
−+=
2
36.99 ft
23815 (23815) 4(266.7)(515827)
52.3 ft
(2)(266.7)
d−
= =
Discard
52.3 ft
(since the cord acts in compression when rebound occurs).
37.0 ftd=
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PROBLEM 13.63
It is shown in mechanics of materials that the stiffness of an elastic cable is k = AE/L
where A is the cross sectional area of the cable, E is the modulus of elasticity and L is
the length of the cable. A winch is lowering a 4000–lb piece of machinery using at a
constant speed of 3ft/s when the winch suddenly stops. Knowing that the steel cable
has a diameter of 0.4 in., E = 29 × 106 lb/in2, and when the winch stops L = 30 ft,
determine the maximum downward displacement of the piece of machinery from the
point it was when the winch stopped.
PROBLEM 13.63 (Continued)
Let position 2 be the position of maximum downward displacement. Let x2 be the elongation in this position.
2
2 2 21
1()
2
32
2 22
32 2
1(121.47 10 ) (4000)( 0.03293)
2
60.735 10 4000 131.72
V xx
xx
= ×− −
= ×− +
Kinetic energy:
22
0 (since 0)Tv= =
Principle of work and energy:
112 2
TVTV+=+
32
22
32
22
2
558.99 65.860 60.735 10 4000 131.72
60.735 10 4000 493.13 0
0.12887 ft
xx
xx
x
+ = ×− +
×− − =
=
Maximum displacement:
21
0.09594 ftxx
δ
=−=
1.151 in.
δ
=
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PROBLEM 13.64
A 2–kg collar is attached to a spring and slides without friction in
a vertical plane along the curved rod ABC. The spring is
undeformed when the collar is at C and its constant is 600 N/m. If
the collar is released at A with no initial velocity, determine its
velocity (a) as it passes through B, (b) as it reaches C.
PROBLEM 13.64 (Continued)
(b) Velocity as the collar reaches C.
PROBLEM 13.65
A 500–g collar can slide without friction along the semicircular rod BCD.
The spring is of constant 320 N/m and its undeformed length is 200 mm.
Knowing that the collar is released form rest at B, determine (a) the speed
of the collar as it passes through C, (b) the force exerted by the rod on the
collar at C.
PROBLEM 13.66
A thin circular rod is supported in a vertical plane by a bracket at A.
Attached to the bracket and loosely wound around the rod is a spring
of constant
3 lb/ft
k=
and undeformed length equal to the arc of
circle AB. An 8–oz collar C, not attached to the spring, can slide
without friction along the rod. Knowing that the collar is released
from rest at an angle
θ
with the vertical, determine (a) the smallest
value of
θ
for which the collar will pass through D and reach Point A,
(b) the velocity of the collar as it reaches Point A.
PROBLEM 13.66 (Continued)
(b) Velocity at A:
Point D:
0 0 1 lb ft[see Part ( )]
DDD
VTV a= = = ⋅