We have observed that the work done against a gravitational or an elastic force depends only on the net change of position and not on the particular path followed in reaching the new position. Forces with this characteristic are associated writh conservative force fields, wrhich possess an important mathematical property.
Consider a force field where the force F is a function of the coordi---?2 nates, Fig. 3/10. The work done by F during a displacement dr of its
point of application is dU — F * dr. The total work done along its path from 1 to 2 is
/ F
dry JJ = I F-dr = [ Wx dx + Fv dy + Fz dz)
r The integral F - d r is a line integral which depends, in general, on the f
1« particular path followed between any two points 1 and 2 in space. If, however, F 'dr is an exact differentialT —dVof some scalar function V of
~~ the coordinates, then
~~~ x
¡•V,
Figure 3/10 17,.a = ~dV = ~(V2 - V^ (3/23)
JV,
which depends oidy on the end points of the motion and which is thus independent of the path followed. The minus sign before dV is arbitrary but is chosen to agree with the customaiy designation of the sign of po-tential energy change in the gravity field of the earth.
If V exists, the differential change in V becomes
dX dy dZ
* Optional.
'Recall that a function dtb = P dx + Q dy + R dz is an exact differential in the coordinates x-y-z if
cty dx Hz fix dz dy
Article 3/7 Potential Energy 181
Comparison with ~dV = F • dr = F,: dx + Fv dy + F, dz gives us
F = F F =
T fix y dy * dz
The force may also be written as the vector
(3/24) where the symbol V stands for the vector operator "del", which is
dx dy c)z
The quantity V is known as the potential function, and the expression VV is known as the gradient of the potential function.
When force components are derivable from a potential as described, the force is said to be conservative, and the work done by F between any two points is independent of the path followed.
182 Chapter S Kinetics of Particles
S a m p l e P r o b l e m 3 / 1 6
The 6-lb slider is released from rest at position 1 and slides with negligible friction in a vertical plane along the circular rod. The attached spring has a stiff-ness of 2 lb/in. and has an unstretched length of 24 in. Determine the velocity of the slider as it passes position 2.
Solution. The work done by the weight and the spring force on the slider will be treated using potential-energy methods. The reaction of the rod on the slider is normal to the motion and does no work. Hence. U[_.z 0. We define the datum to be at the level of position 1, so that the gravitational potential energies are
V1 = 0
Vz= -mgh = - —12 ft-lb The initial and final elastic (spring) potential energies are
Yj = = I ( 2 ) ( 1 2 ) ^ )J - 48 ft-lb
V, = \kx* = ^ ( 2 ) ( 1 2 ) (2^2 - = 8.24 ft-lb Substitution into the alternative work-energy equation yields
[T, + V1 + V[.2 = r2 I VK| 0 + 48 + 0 =
\ ( ¿ J
^ - 12 + 8.24(T) Note that if we evaluated the work done by the spring force acting on the slider by means of the integral J'F * dv, it woidd necessitate a lengthy computation to account for the change in the magnitude of the force, along with the change in the angle between the force and the tangent to the path. Note further that de-pends only on the end conditions of the motion and does not require knowledge of the shape of the path.
S a m p l e P r o b l e m 3 / 1 7
The 10-kg slider moves with negligible friction up the inclined guide. The attached spring has a stiffness of 60 N/m and is stretched 0.6 m in position A, where the slider is released from rest. The 250-N force is constant and the pulley offers negligible resistance to the motion of the cord. Calculate the velocity vc of the slider as it passes point C.
Solution. The slider and inextensible cord together with the attached spring will be analyzed as a system, which permits the use of Eq. 3/2la. The only non-potential force doing work on this system is the 250-N tension applied to the cord. While the slider moves from A to C, the point of application of the 250-N force moves a distance of AB — BC or 1.5 — 0.9 0.6 m.
U'A_C 250(0.6) = 150 J
We define a datum at position A so that the initial and final gravitational poten-tial energies are
V^ = 0 Vc = mgh = 10(9.81)(1.2 sin 30°) = 58.9 J The initial and final elastic potential energies are
VA = \ k x / = | (60K0.S)2 = 10.8 J
^c 2 kxB2 = 2^0(0.6 + 1.2)a = 97.2 J Substitution into the alternative work-energy equation 3/21a gives
fta + va+ ua-c = Tc + vcJ o + 0 - 10.8 + 150 = \ (10)i;ca + 58.9 + 97.2
vc = 0.974 m/s Aijs.
Helpful Hints
(Î) Do not hesitate to use subscripts tai-lored to the problem at hand. Here we use A and C rat her than 1 and 2.
(2) The reactions of the guides on the slider are normal to the direction of motion and do no work.
Article 3/7 P o t e n t i a l Energy 1 8 3
S a m p l e P r o b l e m 3 / 1 8
The system shown is released from rest with the lightweight slender bar OA in the vertical position shown. The torsional spring at O is undeflected in the initial position and exerts a restoring moment of magnitude k„8 on the bar, where 0 is the counterclockwise angular deflection of the bar. The string S is attached to point C of the bar and slips without friction thr ough a vertical hole in the support surface.
For the values mA 2 kg, mB 4 kg, L 0,5 m, and k„ 13N-m/rad:
(а) Determine the speed vA of particle A when 9 reaches 90°.
(б) Plot uA as a function of I) over the range 0 £ 9 £ 90°. Identify the maximum value of vA and the value of 8 at which this maximum occurs.
Solution (a). We begin by establishing a general relationship for the potential energy associated with the deflection of a torsional spring. Recalling that the change in potential energy is the work done on the spring to deform it, we write
We also need to establish the relationship between uA and vB when 8 90°. Not-ing that the speed of point C is always vA!2, and further noting that the speed of cylinder B is one-half the speed of point C at 6 90°, we conclude that at 9 90°,
^li = 1
Establishing datums at the initial altitudes of bodies A and B, and with state 1 at 9 = 0 and state 2 at 9 = 90°, we write
(b). We leave our definition of the initial state 1 as is, but now redefine state 2 to be associated with an arbitrary value of 8. From the accompanying diagram con-structed for an arbitrary value of 9, we see that the speed of cylinder B can be written as
Upon substitution of the given quantities, we vary 9 to produce the plot of vA
versus 8. The maximum value of vA is seen to be
= 1-400 m/s at 9 = 56.4° Ans.
c
Helpful Hints
(T) Note that mass B will move down-ward by one-half of the length of string initially above the supporting surface. This downward distance is 1 (L
¿ U
v 2J " I T
-(2) The absolute-value signs reflect the fact that i'B is known to be positive.
184 Chapter 3 K i n e t i c s of P a r t i c l e s
3/149 The 1.2-kg slider is released from rest in position A and slides without friction along the vertical-plane guide shown. Determine ¡.a.) the speed vB of the slider as it passes position B and (b) the maximum deflection 3 of the spring.
Ans. (a) vB = 9.40 m/s, (b) S = 54.2 m m
PROBLEMS
Introductory Problems
3/147 The spring has an unstretched length of 0.4 m and a stiffness of 200 N/m. The 3-kg slider and attached spring are released from rest at A and move in the vertical plane. Calculate the velocity v of the slider as it l eaches B in the absence of friction.
Ans. v = 1.537 m/s
Problem 3/148
Problem 3/149
3/150 The 1.2-kg. slider of the system of Prob. 3/149 is re-leased from rest in position A and slides without friction along the vertical-plane guide. Determine the normal force exerted by the guide on the slider (a) just before it passes point C, (b) just after it passes point C, and Ic) just before it passes point E.
3/151 The 2-kg plunger is released from rest in the position shown where the spring of stiffness k - 500 N/m has been compressed to one-half its uncompressed length of 200 mm. Calculate the maximum height /; above the starting position reached by the plunger.
Alls, h 95.6 mm Problem 3/147
3/148 The two particles of equal mass are joined by a rod of negligible mass. If they are released from rest in the position shown and slide on the smooth guide in the vertical plane, calculate their velocity v when A leaches B's position and B is at B'.
1.5 m v
P r o b l e m 3/151
Article 3/7 P r o b l e m s 1 8 5
3 / 1 5 2 A bead with a mass of 0.25 kg is released from rest at A and slides down and around the fixed smooth wire. Determine the force N between the wire and the bead as it passes point B.
A
0.6 m
Problem 3/152
3/153 The spring of constant k is unstretched when the slider of mass in passes position B. If the slider is released from rest in position A, determine its speed as it passes points B and C. What is the nor-mal force exerted by the guide on the slider at posi-tion C? Neglect fricposi-tion between the mass and the circular guide, which lies in a vertical plane.
Aras. Lfg
VC
N - I
= mj^
+ (3 - 2m V'2)
4 ^ + ^ ( 3 - 2 / 2 )
kR £3 - 2J2)
3/154 The system is released from rest with the spring initially stretched 3 in. Calculate the velocity v of the cylinder after it has dropped 0.5 in. The spring has a stiffness of 6 lb/in. Neglect the mass of the small pulley.
k = 6 lb/in.
100 lb
Problem 3/1S4
3/155 The light rod is pivoted at O and carries the 5- and 10-Ib particles. If the rod is released from rest at 9 = 60° and swings in the vertical plane, calculate ia) the velocity V of the 5-Ib particle just before it hits the spring in the dashed position and (6) the maximum compression x of the spring. Assume that x is small so that the posi-tion of the rod when the spring is compressed is essentially horizontal.
Arts, (a) v 3.84 ft/sec, f6) A' = 0.510 in.
101b
k = 200 Ib/m.
Problem 3/153
1 8 6 Chapter 3 K i n e t i c s of P a r t i c l e s
Representative Problems
3 / 1 5 6 The 10-kg collar slides on the smooth vertical rod and has a velocity iij - 2 m/s in position A where each spring is stretched 0.1 m. Calculate the veloc-ity v2 of the collar as it passes point B.
0.3 m
3 / 1 5 7 The two wheels consisting of hoops and spokes of negligible mass rotate about their respective cen-ters and are pressed together sufficiently to pre-vent any slipping. The 3-lb and 2-lb eccentric masses are mounted on the rims of the wheels. If the wheels are given a slight nudge from rest in the equilibrium positions shown, compute the angular velocity ft of the larger of the two wheels when it has revolved through a quarter of a revolution and put the eccentric masses in the dashed positions shown. Note that the angular velocity of the small wheel is twice that of the large wheel. Neglect any friction in the wheel bearings.
A P I S . ft = 9 . 2 7 rad'sec 3 1b
P r o b l e m 3/1S7
3 / 1 5 8 In the design of an inside loop for an amusement park ride, it is desired to maintain the same cen-tripetal acceleration throughout the loop. Assume negligible loss of energy during the motion and de-termine the radius of curvature p of the path as a function of the height y above the low point A, where the velocity and radius of curvature are L.'u and pc, respectively. For a given value of pa, what is the minimum value of v0 for which the vehicle will not leave the track at the top of the loop?
• P
\
N. s
V . ^
Problem 3/158
3 / 1 5 9 The mechanism shown lies in the vertical plane and is released from rest in the position for which ft 60°. In this position the spring is unstretched.
Calculate the velocity of the 10-lb sphere when ft 90°. The mass of the links is small and may be neglected.
Ans. v 4.28 ft/sec
15"
P r o b l e m 3/159
Article 3/7 P r o b l e m s 187
3/160 The small bodies A and B each of mass m are con-nected and supported by the pivoted links of negli-gible mass. If A is released from rest in the posit ion shown, calculate its velocity vA as it crosses the ver-tical centerline. Neglect any friction.
3/161
Problem 3/160
When the mechanism is released from rest in the position where 0 60°, the 4-kg carriage drops and the 6-kg sphere rises. Determine the velocity t' of the sphere when 8 = 180°. Neglect the mass of the links and treat the sphere as a particle.
Ans. v = 0.990 m/s
6 kg
3 / 1 6 2 The springs are undeformed in the position shown.
If the 14-lb collar is released from rest in the posi-tion where the lower spring is compressed a in., de-termine the maximum compression xB of the upper spring.
kB = 10 lb/in.
ki = 48 lb/in.
Problem 3/162
3 / 1 6 3 If the system is released from rest, determine the speeds of both masses after B has moved 1 m. Ne-glect friction and the masses of the pulleys.
Anil. vA 0.616 m/s, i/g = 0.924 m/s
8kg
Problem 3/163
P r o b l e m 3/161
1 8 8 Chapter 3 K i n e t i c s of P a r t i c l e s
3 / 1 6 4 The system is released from rest in the position shown. The 6-kg cylinder passes through the hole in the bracket, but the 4-kg collar does not. Deter-mine the maximum height h which the S-kg cylin-der rises. Explain what happens to the kinetic energy of the collar. Neglect the mass of the cable and small pulleys.
Problem 3/164
3/165 A satellite is put into an elliptical orbit around the earth and has a velocity vp at the perigee position P. Determine the expression for the velocity at the apogee position A. The radii to A and P are, spectively, and ;>. Note that the total energy re-mains constant.
'> f p F i
Problem 3/165
3 / 1 6 6 The collar has a mass of 2 kg and is attached to the light spring, which has a stiffness of 30 N/m and an unstretched length of 1.5 m. The collar is released from rest at A and slides up the smooth rod under the action of the constant 50-N force. Calculate the velocity v of the collar as it passes position B.
bV
1.5 m 30=
SO N V
1-1 I--J I I 1-1
t—r ı—i )—I i
-I r-1 k-30 N/m
A A / W W W W V W H
2 m
Problem 3/166
3 / 1 6 7 Upon its return voyage from a space mission, the spacecraft has a velocity of 24 000 km/h at point A, which is 7000 km from the center of the eaith. De-termine the velocity of the spacecraft when it reaches point B, which is 6500 km from the center of the earth. The trajectory between these two points is outside the effect of the earth's atmosphere.
Aits. I'B = 26 300 km/h
5 * (
P r o b l e m 3/167
Article 3/10 P r o b l e m s 189
3 / 1 6 8 The 5-Ib sliding collar C with attached spring moves with friction from A to B along the fixed rod.
If the collar has a velocity of 6 ft/sec at A and a ve-locity of 10 ft/sec at B, determine the loss Uf of en-ergy due to friction. The spring has a stiffness of 2 lb/ft and an unstretched length of 2 ft. The jc-y plane is horizontal Also determine the average frict ion force F during the mot ion from A to B.
z
3 / 1 6 9 The fixed point O is located at one of the two foci of the elliptical guide. The spring has a stiffness of 3 N/m and is unstretched when the slider is at A. If the speed is such that the speed of the 0.4-kg slider approaches zero at C, determine its speed at point B. The smooth guide lies in a horizontal plane. (If necessary, refer to Eqs. 3/43 for elliptical geometry.)
Ans. = 2.51 m/s
B
3 / 1 7 0 A spacecraft in is heading toward the center of the moon with a velocity of 2000 mi/hr at a distance from the moon's surface equal to the radius R of the moon. Compute the impact velocity u with the sur-face of the moon if the spacecraft is unable to fire its retro-rockets. Consider the moon fixed in space. The radius R of the moon is 1080 mi, and the accelera-tion due to gravity at its surface is 5.32 ft/sec2.
R J v j
1
I
Problem 3/170
3/171 A 175-lb pole vaulter carrying a uniform 16-ft, 10-lb pole approaches the jump with a velocity v and man-ages to barely clear the bar set at a height of 18 ft.
As he clears the bar, his velocity and that of the pole arc essentially zero. Calculate the minimum possible value of v required for him to make the jump. Both the horizontal pole and the center of gravity of the vaulter arc 42 in. above the ground during the approach.
Ans. v 20.4 mi/hr
Problem 3/171
800 mm
Problem 3/169
190 Chapter J K i n e t i c s of P a r t i c l e s
3/172 When the 10-lb plunger is released from rest in its vertical guide at $ 0, each spring of stiffness k = 20 lb/in. is uncompressed. The links are free to slide through their pivoted collars and com-press their springs. Calculate the velocity v of the plunger when the position 0 - 30° is passed.
Problem J/172
3/173 The car's of an amusement-park ride have a speed if[ = 90 km/h at the lowest part of the track. Deter-mine their speed v2 at the highest part of the track.
Neglect energy loss due to friction. (Caution: Give careful thought to the change in potential energy of the system of cars.)
A»s. v2 • 35.1 km/h
"2
Problem J/173
3 / 1 7 4 An artificial satellite moving in an elliptical orbit has a velocity of 25 000 km/h at an altitude of 2200 km at point A. Determine its velocity va at point B where the altitude is 2500 km. Treat the earth as a sphere of radius if 6371 km aird use g 9.825 m/s2 for the acceleration of gravity at the earth's surface.
\
Problem J/174
3 / 1 7 5 The 0.6-kg slider is released from rest at A and slides down the smooth parabolic guide (which lies in a vertical plane) under the influence of its own weight and of the spring of constant 120 N/m. De-termine the speed of the slider as it passes point B and the corresponding normal force exerted on it by the guide. The unstretched length of the spring is 200 mm.
Arts. vB 5.92 m/s, iV" = 84.1 N
P r o b l e m J/175
Article 3/7 P r o b l e m s 191
3 / 1 7 6 Ail instrument package of mass m is attached at A to two springs each of stiffness k and unstretched length b. The package is released from rest at this point and falls a short distance. The deflection y at any instant of time is very small compared with b, so that the stretch of the spring is very nearly given by .r = y sin Q where 0 = cos 1 (c/'b). Determine the velocity y of the package as a funct ion of y and find the maximum deflection yma): of the package.
3 / 1 7 7 Calculate the maximum velocity of slider B if the system is released from rest with x y. Motion is in the vertical plane. Assume friction is negligible.
The sliders have equal masses.
Ans. (ygJniaj, = 0.962 m/s
Problem 3/177
3 / 1 7 8 By "pumping" as he swings, the boy increases the swing amplitude from 80 to 81 by abruptly changing from the sitting to the supine position at the start of each forward swing and reversing positions at the start of each back swing. Treat the boy as a particle in each of the two configurations where the path of his mass center is shown by the dashed tra-jectory. Any loss of mechanical energy (T + Vg) be-tween Gj and G;i may be assumed to be negligible.
Express $i in terms of 8a, R, and h.
A
Problem 3/178
• 3 / 1 7 9 The chain starts from rest with a sufficient number of links hanging over the edge to barely initiate mo-tion in overcoming fricmo-tion between the remainder of the chain and the horizontal supporting surface.
Determine the velocity v of the chain as the last link leaves the edge. The coefficient of kinetic fric-tion is p.).. Neglect any fricfric-tion at the edge.
, I gL Ans. v = /
V 1 + ftt
P r o b l e m 3/179