Unlock document.
This document is partially blurred.
Unlock all pages and 1 million more documents.
Get Access
PROBLEM 13.72 (Continued)
PROBLEM 13.73
A 10-lb collar is attached to a spring and slides without
friction along a fixed rod in a vertical plane. The spring
has an undeformed length of 14 in. and a constant k = 4
lb/in. Knowing that the collar is released from rest in the
position shown, determine the force exerted by the rod on
the collar at (a) Point A, (b) Point B. Both these points are
on the curved portion of the rod.
SOLUTION
2
10 0.31056 lb s /ft
W
PROBLEM 13.73 (Continued)
B
B
PROBLEM 13.74
An 8-oz package is projected upward with a
velocity v0 by a spring at A; it moves around a
frictionless loop and is deposited at C. For each of
the two loops shown, determine (a) the smallest
velocity v0 for which the package will reach C, (b)
the corresponding force exerted by the package on
the loop just before the package leaves the loop at
C.
SOLUTION
(
a) The smallest velocity at B will occur when the force exerted by the
C
PROBLEM 13.74 (Continued)
(
b)
C
PROBLEM 13.75
If the package of Problem 13.74 is not to hit the
horizontal surface at C with a speed greater than
10 ft/s, (a) show that this requirement can be
satisfied only by the second loop, (b) determine
the largest allowable initial velocity
0
v
when the
second loop is used.
SOLUTION
0
0
PROBLEM 13.76
A small package of weight W is projected into a
vertical return loop at A with a velocity v0. The
package travels without friction along a circle
of radius r and is deposited on a horizontal
surface at C. For each of the two loops shown,
determine (a) the smallest velocity v0 for which
the package will reach the horizontal surface at
C, (b) the corresponding force exerted by the
loop on the package as it passes Point B.
SOLUTION
PROBLEM 13.76 (Continued)
Newton’s second law at position B.
2
33
B
n
v gr
ma m m mg
rr
= = =
PROBLEM 13.77
The 1 kg ball at A is suspended by an inextensible cord and given an initial
horizontal velocity of 5 m/s. If
0.6 ml=
and
0,
B
x=
determine yB so that the
ball will enter the basket.
SOLUTION
02
PROBLEM 13.77 (Continued)
B
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.
SOLUTION
5
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
∂∂
∂∂
∂∂
∂∂∂∂∂∂
= = =
SOLUTION
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
ABCA
PROBLEM 13.82*
The potential function associated with a force P in space is known
to be
2 2 2 1/ 2
(, ,) ( ) .V xyz 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)
OABD OD
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.
OABDO
PROBLEM 13.84*
The force
2 2 2 3/2
( )/( )xyz x y z= ++ + +F i jk
acts on the particle
(, , )Px y z
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.
SOLUTION
2
