Unlock document.
This document is partially blurred.
Unlock all pages and 1 million more documents.
Get Access
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.
2
2
2
2
(266.67)(45 )
ge
Vd
= −
PROBLEM 13.62 (Continued)
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.
SOLUTION
2
4000 124.22 lb s /ft
W
1
2
PROBLEM 13.63 (Continued)
Let position 2 be the position of maximum downward displacement. Let x2 be the elongation in this position.
2
1()
21
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.
SOLUTION
Spring elongations:
At , 250 mm 150 mm 100 mm 0.100 m
A
Ax
=−==
B
B
PROBLEM 13.64 (Continued)
C
C=v
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.
SOLUTION
2
C AC
e
PROBLEM 13.65 (Continued)
309.23 mm 200 mm 109.23 mm 0.10923 m
AC
L∆= − = =
( )
( )( )
2
1320 N/m 0.10923 m 1.90909 J
2
Ce
V= =
BBCC
TV TV+=+
2
0 4.0394 0.25 1.90909
C
v+=+
(a)
2 22
4.0394 1.90909 8.5212 m /s
0.25
C
v−
= =
2.92 m/s
C
v=
Find force of rod on collar AC Free Body Diagram
0
z
F=
(no friction)
xy
FF= +Fij
1
75
tan 14.04
300
θ
−
= = °
( )
( )
cos sin
AC
kL
θθ
=∆+
e
F ik
( )( )( )
320 0.10923 cos14.04 sin14.04= °+ °
e
F ik
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/ftk=
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.
SOLUTION
PROBLEM 13.66 (Continued)
PROBLEM 13.67
Cornhole is a game that requires you to toss beanbags
through a hole in a wooden board. People with limited
arm mobility often have difficulty enjoying this favorite
tailgating activity. An adapted launching device attaches
to a wheelchair so that points O and A are fixed. The
device mimics an underhand throw by utilizing an elastic
band to power the arm OC, which rotates about pin O.
The elastic cord has an unstretched length of 1 ft and is
attached to fixed point A and to point B on the arm. The
combined weight of the beanbag and holder at C is 4 lbs,
and you can neglect the weight of the rod OB. Knowing
that the starting position is 30 degrees from the
horizontal as shown in the figure, determine the spring
constant if the velocity of the bean bag is 31 ft/s when
the bag is released at an angle of θ = 45 degrees.
SOLUTION
PROBLEM 13.67 (Continued)
PROBLEM 13.68
A spring is used to stop a 50-kg package which is moving down a 20º
incline. The spring has a constant
30 kN/mk=
and is held by cables so
that it is initially compressed 50 mm. Knowing that the velocity of the
package is 2 m/s when it is 8 m from the spring and neglecting friction,
determine the maximum additional deformation of the spring in
bringing the package to rest.
SOLUTION
Let position 1 be the starting position 8 m from the end of the spring when it is compressed 50 mm by the
cable. Let position 2 be the position of maximum compression. Let x be the additional compression of the
spring. Use the principle of conservation of energy.
11 2 2
.TV T V+=+
22
11
PROBLEM 13.69
Solve Problem 13.68 assuming the kinetic coefficient of friction between
the package and the incline is 0.2.
PROBLEM 13.68 A spring is used to stop a 50-kg package which is
moving down a 20° incline. The spring has a constant
30 kN/mk=
and is held by cables so that it is initially compressed 50 mm. Knowing
that the velocity of the package is 2 m/s when it is 8 m from the spring
and neglecting friction, determine the maximum additional deformation
of the spring in bringing the package to rest.
SOLUTION
Let position 1 be the starting position 8 m from the end of the spring when it is compressed 50 mm by the
cable. Let position 2 be the position of maximum compression. Let x be the additional compression of the
12
(0.2)(460.92)
92.184
92.184(8 )
737.47 92.184
f
U Fd
x
x
→
=
=
= −
=−+
=−−
PROBLEM 13.69 (Continued)
PROBLEM 13.70
A section of track for a roller coaster consists of two circular
arcs AB and CD joined by a straight portion BC. The radius
of AB is 27 m and the radius of CD is 72 m. The car and its
occupants, of total mass 250 kg, reach Point A with
practically no velocity and then drop freely along the track.
Determine the normal force exerted by the track on the car as
the car reaches point B. Ignore air resistance and rolling
resistance.
SOLUTION
Calculate the speed of the car as it reaches Point B using the principle of conservation of energy as the car
PROBLEM 13.70 (Continued)
PROBLEM 13.71
A section of track for a roller coaster consists of two circular
arcs AB and CD joined by a straight portion BC. The radius of
AB is 27 m and the radius of CD is 72 m. The car and its
occupants, of total mass 250 kg, reach Point A with
practically no velocity and then drop freely along the track.
Determine the maximum and minimum values of the normal
force exerted by the track on the car as the car travels from A
to D. Ignore air resistance and rolling resistance.
PROBLEM 13.71 (Continued)
PROBLEM 13.72
A 1-lb collar is attached to a spring and slides without friction
along a circular rod in a vertical plane. The spring has an
undeformed length of 5 in. and a constant
10 lb/ft.k=
Knowing that the collar is released from being held at A
determine the speed of the collar and the normal force between
the collar and the rod as the collar passes through B.
SOLUTION
2
10.031056 lb s /ft
W
B
B
