PROBLEM 13.78
The pendulum shown is released from rest at A and swings through 90°
before the cord touches the fixed peg B. Determine the smallest value of
a for which the pendulum bob will describe a circle about the peg.
PROBLEM 13.79*
Prove that a force F(x, y, z) is conservative if, and only if, the following relations are satisfied:
yy
xx
zz
FF
FF
FF
yxzyxz
∂∂
∂∂
∂∂
∂∂∂∂∂∂
= = =
PROBLEM 13.80
The force
( )/yz zx xy xyz= ++F ijk
acts on the particle
(,,)Pxyz
which moves in space. (a) Using the
relation derived in Problem 13.79, show that this force is a conservative force. (b) Determine the potential
function associated with F.
PROBLEM 13.80 (Continued)
PROBLEM 13.81*
A force F acts on a particle P(x, y) which moves in the xy plane.
Determine whether F is a conservative force and compute the work of
F when P describes the path ABCA
knowing that
(a)
( ) ( )
,kx y kx y=+++Fij
(b)
( ) ( )
.kx y x ky= + ++Fij
PROBLEM 13.82*
The potential function associated with a force P in space is known
to be
2 2 2 1/2
(,,) ( ) .Vxyz x y z=−++
(a) Determine the x, y, and z
components of P. (b) Calculate the work done by P from O to D
by integrating along the path OABD, and show that it is equal to
the negative of the change in potential from O to D.
PROBLEM 13.82* (Continued)
2 2 1/2
2 2 1/2 0
0
2 2 1/2 2 1/2
( 2)
( 2) (2) (3 2)
aa
BD
BD
OABD O A A B B D
y
ya
U aa a a
U UUU
−−−
+
=+− =−
=++
∫
( 2 1) ( 3 2 )aa a=+ −+ −
3
OABD
Ua=
2221/2
( , , ) (0,0,0)
( )0
OD
V Vaaa V
aaa
∆= −
=− ++ −
3
OD
Va∆=−
Thus,
OABD OD
UV= −∆
PROBLEM 13.83*
(a) Calculate the work done from D to O by the force P of
Problem 13.82 by integrating along the diagonal of the cube.
(b) Using the result obtained and the answer to part b of Problem
13.82, verify that the work done by a conservative force around
the closed path OABDO is zero.
PROBLEM 13.82 The potential function associated with a force
P in space is known to be V(x, y, z)
2 2 2 1/2
( ).xyz=−++
(a) Determine the x, y, and z components of P. (b) Calculate the
work done by P from O to D by integrating along the path OABD,
and show that it is equal to the negative of the change in potential
from O to D.
PROBLEM 13.84*
The force
2 2 2 3/2
( )/( )xyz x y z= ++ + +F i jk
acts on the particle
(, , )Pxyz
which moves in space. (a) Using
the relations derived in Problem 13.79, prove that F is a conservative force. (b) Determine the potential
function V(x, y, z) associated with F.
PROBLEM 13.85
(a) Determine the kinetic energy per unit mass which a missile must have after being fired from the surface of
the earth if it is to reach an infinite distance from the earth. (b) What is the initial velocity of the missile
(called the escape velocity)? Give your answers in SI units and show that the answer to part b is independent
of the firing angle.