Exercício Dinâmica - Engenharia Mecânica (1869)

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649 
 
 
PROBLEM 13.96 
A 1.5-lb ball that can slide on a horizontal frictionless surface is attached to a 
fixed Point O by means of an elastic cord of constant 1 lb/in.k = and 
undeformed length 2 ft. The ball is placed at Point A, 3 ft from O, and given an 
initial velocity 0v perpendicular to OA. Determine (a) the smallest allowable 
value of the initial speed 0v if the cord is not to become slack, (b) the closest 
distance d that the ball will come to Point O if it is given half the initial speed 
found in part a. 
 
SOLUTION 
Let L1 be the initial stretched length of the cord and L2 the length of the closest approach to Point O if the cord 
does not become slack. Let position 1 be the initial state and position 2 be that of closest approach to Point O. 
The only horizontal force acting on the ball is the conservative central force due to the elastic cord. At the point 
of closest approach the velocity of the ball is perpendicular to the cord. 
Conservation of angular momentum: 1 1 2 2
1 0
1 0 2 2 2
2
 or 
r mv r mv
L v
L mv L mv v
L
=
= =
 
Conservation of energy: 1 1 2 2T V T V+ = + 
 2 2 2 2
1 1 0 2 2 0
2 2 2 2
1 2 1 0 2 0
2
2 2 2 21
0 0 1 0 2 02
2
1 1 1 1
( ) ( )
2 2 2 2
[( ) ( ) ]
[( ) ( ) ]
mv k L L mv k L L
k
v v L L L L
m
L k
v v L L L L
mL
+ − = + −
− = − − + −
− = − − + −
 
Data: 0 12 ft, 3 ftL L= = 
 2 0 2 ftL L= = for zero tension in the cord at the point of closest approach. 
 
2
1 lb/in. 12 lb/ft
/ 1.5/32.2 0.04658 lb s /ft
k
m W g
= =
= = = ⋅
 
 
2
2 2 2
0 02
2
0
(3) 12
[(3 2) (2 2) ]
0.04658(2)
1.25 257.6
v v
v
− = − − + −
− = −
 
(a) 2 2 2
0 206.1 ft /sv = 0 14.36 ft/sv = 