Phys 7 HW – Energy Systems – fall 2004 – EJZ
Solutions courtesy of the publisher
Due week 5, Thus.28.Oct.2004
Ch.7 # 2 (meteorite), 3 (proton), 16 (lift), 66 (ski lift)
Ch.8 (p.187), Q5 (U), Q9 (K), #2 (drop), 3 (bowl), 9 (bowl), 10 (drop), 37 (U,K), 112
(Mt.Everest), 121 (hydro), 125 (waterfall),
(Challenge 8.45)
Ch.7 # 2 (meteorite)
2. (a) The change in kinetic energy for the meteorite would be


1
1
K  K f  Ki   Ki   mi vi2   4  106 kg 15  103 m/s
2
2

2
 5  1014 J ,
or | K | 5 1014 J . The negative sign indicates that kinetic energy is lost.
(b) The energy loss in units of megatons of TNT would be
 1 megaton TNT 
K   5  1014 J  
  0.1megaton TNT.
15
 4.2  10 J 
(c) The number of bombs N that the meteorite impact would correspond to is found by
noting that megaton = 1000 kilotons and setting up the ratio:
N
0.1  1000 kiloton TNT
 8.
13kiloton TNT
7.3 (proton)
3. (a) From Table 2-1, we have v 2  v02  2ax . Thus,
v  v02  2ax 
 2.4  10 
7 2
 2  3.6  1015   0.035   2.9  107 m/s.
(b) The initial kinetic energy is
Ki 
2
1 2 1
mv0  1.67  1027 kg  2.4  107 m/s   4.8  10 13 J.
2
2
The final kinetic energy is
Kf 
2
1 2 1
mv  1.67  1027 kg  2.9  107 m/s   6.9  10 13 J.
2
2
The change in kinetic energy is K = (6.9  10–13 – 4.8  10–13) J = 2.1  10–13 J.
Page 1 of 7
7.16 (lift) In both cases, there is no acceleration, so the lifting force is equal to the
weight of the object.
(a) Eq. 7-8 leads to W  F  d  (360 kN) (0.10 m)  36 kJ.
(b) In this case, we find W = (4000 N)(0.050 m)  2.0 10 2 J .
7.66 (ski lift)
66. The total weight is (100 people)(660 N/person) = 6.6  104 N=mgh, and the words
“raises … at constant speed” imply zero acceleration, so the lift-force is equal to the total
weight. Thus the power required by the ski lift is
P = Energy/time = mgh/time = (6.6  104N)(150m/60s) = 1.65  105 W.
Ch.8 Q5 (U), Q9 (K), #2 (drop), 3 (bowl), 9 (bowl), 10 (drop), 37 (U,K), 112
(Mt.Everest), 121 (hydro), 125 (waterfall),
(Challenge question: 8.45)
Ch.8 Q5 (U) For the potential energy diagram in Fig.8-24,
(a) rank regions AB, BC, CD, DE according to the greatest force on the particle
Force = -slope of U(x). Steepest slope = greatest force: AB
CD is next steepest, and BC=DE have zero slope = zero force
What value must the mechanical energy E of the particle not exceed if the particle is to be
(b) trapped in the potential well at the left: E<5J
© trapped in the potential well at the right, and E<5J
(d) able to move between the two potential wells but not to the right of point H? E<6J
For the situation in (d), in which of the regions BC, DE, FG will the particle have
(e) the greatest kinetic energy: at the bottom of the lowest well: FG
(f) the least speed: at the highest point: H (or the equivalent 6J point between AB)
Page 2 of 7
8. Q9 (K) In Fig 8-28, is the block’s K increasing, decreasing, or constant in each
region?
(a) AB: speeding up: K increasing
(b) BC: slowing down: K decreasing
© CD: constant speed: K unchanging
(d) how is the block’s total energy E changing in each region? NO CHANGE in E, if the
ramp is frictionless, since gravity is conservative.
8.2 (drop) see figure above right
2. (a) Noting that the vertical displacement is 10.0 – 1.5 = 8.5 m downward (same
direction as Fg ), Eq. 7-12 yields
Wg  mgd cos   (2.00) (9.8) (8.5) cos 0  167 J.
0
(b) One approach (which is fairly trivial) is to use Eq. 8-1, but we feel it is instructive to
instead calculate this as U where U = mgy (with upwards understood to be the +y
direction).
U  mgy f  mgyi  (2.00) (9.8) (15
. )  (2.00) (9.8) (10.0)  167 J.
(c) In part (b) we used the fact that Ui = mgyi =196 J.
(d) In part (b), we also used the fact Uf = mgyf = 29 J.
(e) The computation of Wg does not use the new information (that U = 100 J at the
ground), so we again obtain Wg = 167 J.
(f) As a result of Eq. 8-1, we must again find U = –Wg = –167 J.
(g) With this new information (that U0 = 100 J where y = 0) we have
Ui = mgyi + U0 = 296 J.
(h) With this new information (that U0 = 100 J where y = 0) we have
Uf = mgyf + U0 = 129 J.
We can check part (f) by subtracting the new Ui from this result.
Page 3 of 7
8.3 (bowl)
(a) The force of gravity is constant, so

 

the work it does is given by W  F  d , where F is the force and d is the displacement.
The force is vertically downward and has magnitude mg, where m is the mass of the
flake, so this reduces to W = mgh, where h is the height from which the flake falls. This is
equal to the radius r of the bowl. Thus
W  mgr  (2.00  103 kg) (9.8 m s2 ) (22.0  102 m)  4.31  103 J.
(b) The force of gravity is conservative, so the change in gravitational potential energy of
the flake-Earth system is the negative of the work done: U = –W = –4.31  10–3 J.
(c) The potential energy when the flake is at the top is greater than when it is at the
bottom by |U|. If U = 0 at the bottom, then U = +4.31  10–3 J at the top.
(d) If U = 0 at the top, then U = – 4.31  10–3 J at the bottom.
(e) All the answers are proportional to the mass of the flake. If the mass is doubled, all
answers are doubled.
8.9 (bowl) (a) If Ki is the kinetic energy of the flake at the edge of the bowl, Kf is its
kinetic energy at the bottom, Ui is the gravitational potential energy of the flake-Earth
system with the flake at the top, and Uf is the gravitational potential energy with it at the
bottom, then Kf + Uf = Ki + Ui.
Taking the potential energy to be zero at the bottom of the bowl, then the potential energy
at the top is Ui = mgr where r = 0.220 m is the radius of the bowl and m is the mass of the
flake. Ki = 0 since the flake starts from rest. Since the problem asks for the speed at the
1
bottom, we write mv 2 for Kf. Energy conservation leads to
2
 
Wg  Fg  d  mgh  mgL (1  cos ) .
The speed is v  2 gr  2.08 m/s .
(b) Since the expression for speed does not contain the mass of the flake, the speed would
be the same, 2.08 m/s, regardless of the mass of the flake.
(c) The final kinetic energy is given by Kf = Ki + Ui – Uf. Since Ki is greater than before,
Kf is greater. This means the final speed of the flake is greater.
Page 4 of 7
8.10 (drop)
10. We use Eq. 8-17, representing the conservation of mechanical energy (which neglects
friction and other dissipative effects).
(a) In the solution to exercise 2 (to which this problem refers), we found Ui = mgyi = 196J
and Uf = mgyf = 29.0 J (assuming the reference position is at the ground). Since Ki = 0 in
this case, we have
0  196  K f  29.0
which gives Kf = 167 J and thus leads to
v
2K f
m

2(167)
 12.9 m s.
2.00
(b) If we proceed algebraically through the calculation in part (a), we find Kf = – U =
mgh where h = yi – yf and is positive-valued. Thus,
v
2K f
m
 2 gh
as we might also have derived from the equations of Table 2-1 (particularly Eq. 2-16).
The fact that the answer is independent of mass means that the answer to part (b) is
identical to that of part (a), i.e., v  12.9 m/s .
(c) If Ki  0 , then we find Kf = mgh + Ki (where Ki is necessarily positive-valued). This
represents a larger value for Kf than in the previous parts, and thus leads to a larger value
for v.
8.37 (U,K)
37. If the particle can reach x=1m, what is the speed there? Mass m=0.90 kg, and at
x0=4.5 m, the speed is v0=7.0 m/s in the (-x) direction.
From Fig. 8-48, we see that at x = 4.5 m, the potential energy is U1 = 15 J. If the speed is
v = 7.0 m/s, then the kinetic energy is K1=mv2/2 = (0.90 kg)(7.0 m/s)2/2 = 22 J. The total
energy is E1 = U 1+ K1 = (15 + 22) = 37 J.
Page 5 of 7
(a) At x = 1.0 m, the potential energy is U2 = 35 J. From energy conservation, we have E
= K+U or K2=E-U=37-35= 2.0 J > 0. This means that the particle can reach there with a
corresponding speed (solve K=1/2 mv2 for v:)
2K2
2(2.0 J)

 2.1 m/s.
m
0.90 kg
v2 
(b) The force acting on the particle is related to the potential energy by the negative of the
slope:
Fx  
From the figure we have Fx  
U
x
35  15
  10 N .
24
(c) Since the magnitude Fx  0 , the force points in the +x direction.
(d) At x = 7.0m, the potential energy is U3 = 45 J which exceeds the initial total energy
E1. Thus, the particle can never reach there. At the turning point, the kinetic energy is
zero. Between x = 5 and 6 m, the potential energy is given by
U ( x)  15  30( x  5),
5  x  6.
Thus, the turning point is found by solving 37  15  30( x  5) , which yields x = 5.7 m.
(e) At x =5.0 m, the force acting on the particle is given in the back of the text as
U
(45  15) J
Fx  

  30 N
x
(6  5) m
But this is not entirely accurate. The slope of the potential energy is not well-defined at
this point, since U is flat to the left of 5 and has a positive slope to the right of 5. So it
would be just as true to say the force is zero at this point, since U/x=0. Strictly
speaking, the force is undefined at x=5.
(f) The fact that Fx  0 indicated that the force points in the –x direction. That’s what the
answer in back says. But again, since the slope is not well-defined at this point, the
answer is back is misleading. You could just as well say the force is zero at x=5, and that
force would not be in either direction. Strictly speaking, the direction of the force is
undetermined at x=5.
8.112 (Mt.Everest) (a) The (internal) energy the climber must convert to gravitational
potential energy is U  mgh  90 9.8 8850  7.8  106 J .
b gb gb g
(b) The number of candy bars this corresponds to is
N
7.8 106 J
 6.2 bars.
1.25 106 J bar
Page 6 of 7
8.121 (hydro)
121. We want to convert (at least in theory) the water that falls through h = 500 m into
electrical energy. The problem indicates that in one year, a volume of water equal to Az
lands in the form of rain on the country, where A = 8  1012 m2 and z = 0.75 m.
Multiplying this volume by the density  = 1000 kg/m3 leads to
b gc
hb g
mtotal  Az  1000 8  1012 0.75  6  1015 kg
for the mass of rainwater. One-third of this “falls” to the ocean, so it is m = 2  1015 kg
that we want to use in computing the gravitational potential energy mgh (which will turn
into electrical energy during the year). Since a year is equivalent to 3.2  107 s, we obtain
c2  10 hb9.8gb500g 31.  10

15
Pavg
11
3.2  10
7
W.
8.125 (waterfall)
125. The power generation (assumed constant, so average power is the same as
instantaneous power) is
P
mgh (3/ 4)(1200 m3 )(103 kg / m3 )(9.8m / s 2 )(100 m)

 8.80 108 W.
t
1.0s
Challenge 8.45
45. (a) We take the initial gravitational potential energy to be Ui = 0. Then the final
gravitational potential energy is Uf = –mgL, where L is the length of the tree. The change
is
d
i
U f  U i  mgL  (25 kg) 9.8 m s (12 m)  2.9  103 J .
(b) The kinetic energy is K 
2
1
1
mv 2  (25 kg)(5.6 m/s) 2 =3.9 10 2 J .
2
2
(c) The changes in the mechanical and thermal energies must sum to zero. The change in
thermal energy is Eth = fL, where f is the magnitude of the average frictional force;
therefore,
f 
K  U
3.9 102 J  2.9 103 J

 2.1102 N
L
12 m
Page 7 of 7