Work and Energy Problems
. Below are shown three frictionless tracks. A block is released from rest at the position shown
on the left. To which point does the block make it on the right before reversing direction and
sliding back? Point B is the same height as the starting position.
30.
Ma.\~it 10 _,~
M:l1:cs it to _
.. "
e,_
M:llc.csitlo
31. A spring gun shoots out a plastic ball at speed vo' The spring is then compressed
first shot.
e.. ~ a. ,By what factor is the spring's potential energy increased?
)I-
'f
if
r. Us -::i
b.
iI.
=
twice the distance it was on the
l-f
By want factor is the ball's launch speed increased? Explain.
~)..
~i""'7\JS -:.; ¥,ul>
~
'J,.dr
~
U 0< Y.'2
¥- y. '2.""j
.~
~
lAs -=
V
I-f :: ¥-
C/..
\/7-
'--I 0(. y '2-
~
t--
\/=2
32. A spring is mounted horizontally as shown to the right. A crate, which has a mass of 8.5 kg is pressed against the
spring with a force of 350 N. As a result the spring is compressed a distance of 82.0 cm. The mass is then released
and is allowed to slide along the horizontal, frictionless surface.
a. What is the spring constant of this spring?
K ~ ~i
:0
F.
::\:~1.JD.~J}\
~
~
~
~.
.
~JUt~
rn~.
.. =
.
b. How much elastic potential energy will be stored in the spring?
US ~ ~ ¥.'1--I
~
~\-'-,
-l-f-"3-. S-1"-)
(4 '2.li. ll')( •8J-)?
d. What will be the kinetic energy of this crate after it has left the spring?
f(.'~
c.b~r U \( -05\
1. '"
1
'11.."('{\ V
-;::
lJ.!) ::: ~ t.n. S
~
]
e. What will be the velocity of the crate after it has left the spring?
If. -:: ~
r-r. V'2.
::.')
-= i('(.~
y 12~!I.C1= (~.S)Y
\ l.-f~S
Suppose now that there is friction between the crate and the horizontal surfac an t at t e coe icient of kinetic
friction between the crate and the surface is ).l = 0.65. As a result the crate slows down until it stops.
f. What will be the magnitude of the frictional force acting on the crate as it slides across the surface~.
~
r-
c;.. - (",.",)(q.~)~
1"N='J - \..!
'6~.~
1'1
(
~
\
5~:::ftJ'.N :::,-:V5)(n.3)=r,/SN
.
+."
Sl'-~
g. How much work will be done on the crate by the frictional force from the time the crate is released until ~e mass
stops?
1/
\No;:
OJ( -= Kf - f-.l
o-
~e_
r
v~ :- 0 \0/ (.
;l
V.-t3.~ = ~\4,.,,:>:r J
h. How far will the mass slide before it comes to a halt?
on-
CDcW1
to
0--
~
'oD;<.
c.o'fV"\P\~k
~y.
1.. . L6 \'YI
\
11
+)<;
Work and Energy Problems
33. A 2.45 kg rock is dropped from the top of a 15.5 m vertical cliff.
a.
What is the potential energy of the rock relative to the base of the cliff b~fore it is dropped?
lA~=
b.
yY\~
:=
What is the kinetic energy of the rock just before it hits the ground below?
u.~::~ =-
~~ = c:~ '1lo
c.
\"'1.2../.5'J' \
What is the velocity of the rock at a point 5.00 m above the base of the cliff?
Ua
d.
(1.4'5XQ.l5)(IS.s) = \3lZ.ISJ"]
l
i=
-=-
-t \(. =) \(:: U~i
- UxS)
U,.q('!i)
(]
r01~(t>\tI.') ~ {r/W 'l.
.K')(0-15.5)=
How much total energy does the rock have half way down.
h>WJ. ~
~
\\-\Q S~
~y
7-
v"2=
2.(102,'1)
~ -=
1<1.~S ~ 1
~., 12.1'5
~lPltu;l-G
J ).
34. A roller coaster is at the top o~ j 75.0 In hill. It rolls down the hill o~ very low friction wheels and climbs up to the
top of a 45.0 m hill. Find: (aJ the speed of the thing at the bottom of the first hill
\,.;=,$'
v=1.
K
('1.'3')(,5) = ~ ,,7..
r~"~)
= 11"" 1..
13'5 :: i
u.~c;) '::
~l"Lj,
~1
\/'2.
(b) its speed at the top of the second hill.
.- E
t~-
~COll"')
f _ U.
=
('i.q){1'5)"
i'r.
•..."\
.l. """
If.,,) -+ "2..'r'"
(Gf.r)ll-l S) •. "L" 'l.
1)"<'1
(
\"
'L
~
,,'1.=
\Ljl0~
,,-=
":!>i. ~
~'14 ::~y7...
,,"1-=
U'("l")~
q"<S)
13'5::YYI-t.\""J.
.
Y=2ttro.
35. A roller coaster starts at some height that you do not know. It goes down this hill and the
s up a second hill
that is 28.5 m above the first drop at a speed of22.5 m/s. $0 how high was the initial hill?
.
2.''2
•.-C::;
\'
.l-
",."> ~
' ')..
\l'?
.r'
--I 'i.
i: = \J.o. '1.' .';J + Il/t--
~
l'f.cth,. -=
~s
J l '(
Y'yI.. ~ C'2?,. S )
'fqb'h -=-('l.V)('28'.S)~
36. A roller coaster sits at the
height of 86.0 meters, the
at the bottom of the hill.
a. What is the gravitational
coaster while at the top
.I..
7..
'2. """ V
.
-\
"I-
~("Z"2..C;;)
J
h = '2., (L"3 -+ S " .
.•
q '1:h -=. ":> '02 . '-I '3
"I
••
'~---=-:-7'""---=-=-~\
\h
- SL.t.
1>"0m
J
top of a hill and is preparing to enter a loop-the-Icop as shown to the right. The hill has a
loop has a radius of 22.0 meters and the roller coaster has a mass of 525 kg. Assume h=o
potentiai energy of the roller
of the hill?
lA} -= ~\-.>i "- C?'25) ('1.8')( '6~)= L.H'2.Yl0 ;:s
A
b. What will be the velocity of the roller coaster when it
reaches point C?
\
1.
~~K~~~~m~
.
H4;410= ~('5'l.S),,'L
1/1..-:., ~ lJ>.S').lP
\
f,;: -q1.i~_J
c. What will be the velocity of the roller coaster when it
reachespointA?
~411.j1()
::(?'L'5)l'l.r)(44)
5c~~+
14'1'2.."110::'2'2..1,'01'0
tArt'-:: l).~,.l' 'f(
'lll.'o"lo=
-I-
+1.«;,1.5)"
1..~L.5v''1.
2.,V2. '5 V 'l.~
'L
=
V?
IV
-=
d. What will be the velocity of the roller coaster when it reaches poin~ B?
lAa.; - U.
- ~. -
44'2"\10
¥-
~'b t
:: l02.6'X"l.")b
/"
1..
)
.."2~ -I ~(61.'51" -=-7
~2.3.2
7...«.1 '%
J
7.-
1..\47.410'" \ \ n\"lO +?lP'2.6Y
~ 1.'\ '2'(0 ~ '2.(,'2..5 v'"
\['1..". \ 1..S'--\.'.t
\----1
'0=---1\'
~S.L; 5
2
17...5
Work and Energy Problems
37. A pendulum is pulled backward, and is released from a position 30-cm above the lowest point of its
path. Howfast will the 250-g pendulum be traveling when it reaches the lowest point in its swing?
V\,:' .'?> '('f\
(Y':' . 2.6¥-j
y-:.
'?
2.9Lf
= iVt..
is. '(8"
:. y'2.-
\
~ :::q.l{~,.
V:::
2.'-+ '2
iJ
...
• ..
...........•..~
, ,
, .
.
38. in order to thwart burglars, a clever child ties paint cans to 3.0-m long strings, and hangs them from the stairway of
his house. Ifthe 8-kg can of paint is released at an angle of 35', how fast will it be traveling when it hits the burglar
in th face? (assume that the burglar is standing at the lowest point of the can's swing).
v'7-.:- 10. 'S"li'L(
(
\0\": '3 -)(
1(";:" 3(053'0'
Gi :::.
4
-d"
-I' ~.I~
39.
;
A
f...q"
eP
n-=
X
3-x
#
J..
n":
S ,'2.q"2 =-
6' if "'"
--
2-
:i.v
'\(1'
",----J~
::d
A small cube of mass m slides back and forth in a frictionless,
hemispherical bowl of radius R. Suppose the cube is released at
angle O. What is the cube's speed at the bottom of the bowl?
a. Begin by drawing a before-and-after visual overview. Let the
cube's initial position and speed be Yi and v;. Use a similar
notation for the final position and speed.
i----------- ---------
1/ "Z-
2. \
03-1-.4<1
\-\ -:.
r
f::~''2S~]
d
C~.•.f;"') 0 2" ~
l'
)C~ '2.y'51
".70'-
o
----- ..-------.
!
~. At the initial position, arc ci;hcr K; or (Ug)i zero? If so, which? __
c. At the final position, are either Kr or(Uglr zero? If so, which? --
00-
~~_9..
u'5 ~. 0
d. Write the conservation of energy equations in terms of position and speed variables, omitting any
terms that are zero,
e. Substitute into your expression in d above the appropriate expression of y in terms ofR and
')(-:::tLCObt7
&
~
x.
V-f- ::.
e.
J 2 9 (e- Rem ~ I
hi
13
Power Problems
40. A 100-kg cheetah moves from rest to 30 m/s in 4 s. What is the power?
V-::: ~
t*
.".0-
=
h -'f.~
_
6~
-
b-\:
"2-
{(IOOj(;;oj - :i(IOO)(O)
-==
Y
\7:::
4SDOO
Lt
-
41. A 48.0 kg telephone repairperson climbs up one of them power pole deals. She is carrying 7.85 kg of tools and
things. If she generates 0.765 hp, how much time does it take her to climb the 3.20 m tall pole? (1 hp=746 W)
\/-::: .Y:!-. ~ bUq
(>\,
.11,5
\-t.,<;)C
'S'O. ""Cf - U'3f"- ~~
-::"/
b-t
"=~og.-tlS5:X"1.~)(i...
2)
Dt
o.v~"=. 115\.45\..,
1YIIW -::~l().(P'l'W
I~
')Y
6,O.t..'1
=)
/)t
'51
6 -t:.
~
-=
-==
:: 11'5\.'-1'5"10
) Dt
:'0'0 .lP'j
t bt:;::
1> .Di $' \
42. I left a 150W bulb on for 2.5 hours. Determine how much electricity I used. In this case the electricity (electrical
energy) is being changed into
and Iight...that's the wo~e!
O\,,-e
W")
'0
W
__
)
r-=~
1-.'? n( ~ -;"OOS
ts
.j;,....
1150::..xY-
qooo
w:::6'5o)LC)DOO)~
\;315 b)CC) 0
.:::r
oy
-= 90005
\ .~5X\Ol#J
\ •••••
y
43. A1QQ'!j motor is being used to lift shingles at a constant velocity to the top of a roof. If one pack of shingles has a
28 kg mass, determine the velocity that the pack will be raised at. If the shingles are being raised at a constant
v~then
CVf\S1-o...r+
the net force acting on the pack is?
-.-l--------.
- -RcL( -=
I
~V
t,f
".eloc:l~ -=.' 0
~~
_-b"
V\.R.t"
.ft:..• c..e.
r5-:::.
f
l"'\i:l\1:>,
"y~
(R,rw,M \?t(
\N -=
?-=
~
~a
~v'
?pO ::
Y::'~
~fJ'o ::
---
LJ.
'2./4.\
li~)l'1.nV
~.4
V.
'J.,~. Lt
\j-=.13~
14