Exercício Dinâmica - Engenharia Mecânica (100)

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820 
 
PROBLEM 13.193 
A 60-g steel sphere attached to a 200-mm cord can swing about 
Point O in a vertical plane. It is subjected to its own weight and to 
a force F exerted by a small magnet embedded in the ground. The 
magnitude of that force expressed in newtons is 23000/F r= 
where r is the distance from the magnet to the sphere expressed in 
millimeters. Knowing that the sphere is released from rest at A, 
determine its speed as it passes through Point B. 
 
SOLUTION 
Mass and weight: 0.060 kgm = 
 (0.060)(9.81) 0.5886 NW mg= = = 
Gravitational potential energy: gV Wh= 
where h is the elevation above level at B. 
Potential energy of magnetic force: 
 
2
2
3000
( , in newtons, in mm)
3000 3000
N mm
r
m
dV
F F r
drr
V
rr∞
= = −
= − = ⋅
 
Use conservation of energy: 1 1 2 2T V T V+ = + 
Position 1: (Rest at A.) 
 
1 1
1
1
0 0
100 mm
( ) (0.5886 N)(100 mm) 58.86 N mmg
v T
h
V
= =
=
= = ⋅
 
 From the figure, 
2 2 2 2200 100 (mm )
100 12 112 mm
AD
MD
= −
= + =
 
 
2 22
1
2 2 2
2
1
200 100 112
42544 mm
206.26 mm
r AD MD
r
= +
= − +
=
=
 
 1
1
3000
( ) 14.545 N mmrV
r
= − = − ⋅ 
 3
1 58.86 14.545 44.3015 N mm 44.315 10 N mV −= − = ⋅ = × ⋅