Download 10-16 Energy Homework

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\R~3'MI. The 100 kg box shown ab:~e";~;~:;~~:;~;l~~':;~:~';~e
~~::;:'bY a student. The box slides across
a rough surface, and its position x varies with time t according to the equation x = 0.5e + 2t, where x is in
meters and 1 is in seconds ..
a. Detennine the speed of the box at time 1 = O. (2m/s)
~ \.~-l'L~;
'J= ~~
\ \/~ 2!:f
V ..::-\.SlO) -1-2b. Detennine the following as functions of time I.
i. The kinetic energy of the box
¥-=
II.
[so (\.S/C.'l.+-2t
\-r0.'1L.
1
l
The net force acting on the box (300t)
dv
~dt
I"
-r-:::
iii. The power being delivered to the box (450e+600t)
y::: tv
==
+'2)
(600\:;) ll.5t?
~ ll-f SD~
'3,
-t loot> t
1
'.
c. Calculate the net work done on the box in the interval 1 = 0 to 1 = 2s. (3000 J)
\N -= bY-€ -:: ~ 'rY\(
\if -
%-'.
~ l\oO
y
v( 'L
S("1-)2 +
~r(~~~~i
(,
00) (-eo) ~ \ 1:>coo -:5]
d. Indicate below whether the work done on the box by the student in the interval 1 = 0 to 1 = 2s would be
greater than, less than, or equal to the answer in part (c).
Greater than
__
Less than
__
Equal to
~
your answer
Y
W\ c...~C5Y\ c\\)-e1
o.-e.( '(eQ,y-s ~
Y\~9jA\-\ V R \AI {)'t \(
w....\- \Nay\::-.
\ \).2006M2.
A nonlinear spring is compressed various distances x, and the force F required to compress it is
measured for each distance. The data are shown in the table below.
1_
r:-
1':::
x(m)
X1- l \'V\ 1..')
F(N)
eil,\
~X
,
t-- ~
\/
Us -:: ~f-X
0.05
4
.DDS
0.10
17
. b\
0.15
38
• o"2.'2.S
0.20
68
0.25
106
I .0'-1
.Ou>'2-S
1-
Assul11,ethat the magnitude of the force applied by the spring is of the form F(x) =
Ax' .
a, Which quantities should be graphed in order to yield a straight line whose slope could be used to
calculate a numerical value for A ?
f\ -_
+=
'-t -
7.-
X
b. Calculate values for any of the quantities identified in a. that are not given in the data, and record these
values in the table above. Label the top of the column, including units.
c. On the axeS below, plot the quantities you indicated in (a) . Label the axeS with the variables and
appropriate numbers to indicate the scale.
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, --4----+----1----_:----..j---- ----~---_:- ---i----t----r---'----I
!
.oy,
.02
d.
Using your graph, calculate A.
The spring is then placed horizontally on the floor. One end of the spring is fixed to a wall. A cart of mass 0.50
kg moves on the floor with negligible friction and collides head.on with the free end of the spring, compressing
it a maximum distance of 0.10 m. .
e. Calculate the work done by the cart in compressing the spring 0.10 m from its equilibrium length.
\f'J-=~X
-=(A'/.'l..)(X]
(\I\L1)('\J~
3>
f.
Calculate the speed of the cart just before it strikes
W-::- D¥-C
3
the spring.
~~'T.J
-::-~ U;}v1\I 7-
-:::
1.- .1...?,
l.lI4J~[.5".JJ
(\1:=
\.5\
~
j
-
;
Vo
. ~).
vf- -- 0
.
i;82M2.
A car of mass M moves with an initial speed Vo on a straight horizontal road. The car IS brought to rest by
braking in such a way that the speed of the car is given as a function of time t by v = (va' - Rt/M)x where R is a
constant.
a. Develop an equation that expresses the time rate of change of kinetic energy.
~:::~ ~(Vo~- ~~~
~
c.-c ~~1{~) -=~~
b.
1
Determine the time it takes to bring the car to a complete stop.
D-:::
Vv'l- -
%
\}o1--;: ~
'fY\
Y"A
\l 0'1. ==
'(2.. t.
c.
Develop an equation forthe
acceleration
of the car as a function oftime
d.
On the axes below, sketch the magnitude of the braking force as a function of time t.
Fcrce
r-o
o
I
1
T
t.
T
)R/4
1
E,
D
~----)R----i
\~)
.2011. An amusement park ride features a passenger compal1ment of mass M that is released from rest at point A. as
shown in the figure above, and moves along a track to point E. The compartment is in free fall between points A and
B, which are a distance of 3R!4 apart, then moves along the circular arc of radius R between points Band D.
Assume the track is frictionless from point A to point D and the dimensions of the passenger compartment are
negligible compared to R.
(a) On the dot below that represents the passenger compartment,
draw and label the forces (not components) that
act on the passenger compartment when it is at point C, which is at an angle 8 from point B.
(b)
In terms of 0 and the magnitudes of the forces drawn in part (a), determine an expression for the magnitude of
the centripetal force acting on the compartment at point C. If you need to draw anything besides what you have
shown in part (a) to assist in your solution, use the space below. Do NOT add anything to the figure in part (a).
+c
:;
'Z.\="c..e..0;"..t..<"
~ N - yY\~ ¥\ '(\ e-
(c) Derive an expression
fundamental
constants,
for the speed vD of the passenger
-= .\= c ~
compartment
as it reaches
point D in tenns of M, R, ana
as appropriate.
Tot t:A-= "lot 5p
\
",j.,
0~C~;-1t=) -= i~'Vo
':l (*~~
Y.E.
E
1-
A force acts on the compartment
between points"D- and
an:tbrings it to rest at point E.
(d) If the compartment is brought to rest by friction, calculate the numerical value of the coefficient
between the compartment
and the track.
of friction
Y\I-=- 1::>'E,
(y-r-)lf;QU~I-)-= ~l~:If{)
[,u~- ?i}
(e) Now consider the case in which there is no fi.iction between the compartment
and the track, but instead the
compartment is brought to rest by a braking force-d:1c, where k is a constant and v is the velocity of the
COmpal1ment. Express all algebraic
answers to the following
as appropriate.
i. Write, but do NOT solve, the differential
m c\v
--c\ t.
equation
in terms of
for \'(1).
Ai, R, k, 'D ' and fundamental
II
-T
constants,
p
ii. Solve the differential equation you wrote in part i.
v
VD
e-~
om
-
\'- 'c"
V -:::LV DJe r;..
iii. On the axes below, sketch a graph of the magnitude of the acceleration of the compartment as a function of
time. On the axes, explicitly label any intercepts, asymptotes, maxima, or minima with numerical values or
algebraic expressions, as appropriate.
qgnitu~
ccelerauon
0
TIme
_~t- )
f::YY1o-.-
0-.::=
-~V
dv -- d(vpeyv,
~
- d~
....e--
'('A
_ \"Y\
\/p
~
~
VV\
1=.==D
fA-=:
\/D
~
\4) '2002M3.
An object of mass 0.5 kg experiences a force that is associated with the potential energy
function
U(x)
~,where
=
U is in joules and x is in meters.
2.0+x
a. On the axes below, sketch the graph ofU(x) versus x.
U(J)
X \.A
1-
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0
13
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1
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3.0
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2.0
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b. Determine the force associated with the potential energy function given above.
r
dlA
-r = --d
y...
= - ~ ('2...0 -r-\()
-\
-=) _T
4-
x(m)
5
4
3
2
t
-\
- 4("'.)
- L\ L-' v.:-"2-)
-2. -
C!.o-t X~
c. Suppose that the object is released from rest at the origin. Determine the speed of the particle at x = 2
m.
"\""" xc: =- 2]"" ::-\)., T ¥1;-= \ -t
(.S) V"2\ := ~ (. 5) V 2-
-i
\/=Jy
In the laboratory, you are given a glider of mass 0.50 kg on an air track. The glider is acted on by the
force detennined in part b. Your goal is to detennine experimentally the validity of your theoretical
calculation in part c.
d. From the list below, select the additional equipment you will need from the laboratory to do your
experiment by checking the line to the left of each item. If you need more than one of an item, place the
number you need on the line.
\
Meterstick
Balance
__
Stopwatch
Wood block __
1Photogate
timer
String
___
~Set of objects of different masses
Spring
\ ~~~M2.
A particle of mass m moves in a conservative force field described by the potential energy function U(r) = a(rlb + b/r).
where a and b are positive constants and r is the distance from the origin. The graph of U(r) has the-following shape.
u
Uo --------
o
4"
a.
C\.\- (" S\o\Jt
In terms of the constants a and b, determine the following.
i. The position r, at which the potentiol energy is a minimum
--a- "'"(L
_ d u.
lor -I)
+
_ ,.,
I:J
-r
~ _
ii. The minimum potential
-=
\AO
b.
0-. (
~b
o.r~ =
(}...
( .L
~
0
D
\5
0
-+ :.Q.\
10
-=
("2. )
=>
y-"2-~ b'J.-
0
[('0 - bJ
energy U,
= [CA. 1
T ~)
Sketch the net force on the porticle as a function of r on the graph below, considering a farce directed
from the origin to be positive, and a force directed toward the origin to be negative.
r
___ ~t
F
r
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~
,
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~
,
Jt
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away
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,
---- -----'-----r----,-----j-----r----l-----j-----
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----~-----~-----~----~-----~-----~-----~----~----_.
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----~-----t-----t----~-----i-----t-----~----i-----.
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The particle is released from rest at r = r,/2
v::: 0
c. In terms of U, and m, determine the speed of the particle when it is at r
,E
-=l\,'= (A(~
o(
d.
-'r-Z-) =-
4+2-2-1
1:""-1{
= ~ ""~z.
-= ~
t~.5"-=
=
-+y~
'('n
:}.o...
r, .
-=
v/
Write the equation or equations that could be used to determine where, if ever, th
to rest. It is not necessary to solve for this position.
C\~
toVV\R.S
~
Y'<&"'t" WI,'.JlX'I Ulr)
Briefly and qualitatively
describe
?o...Y'1'1c\<. Sy-e.u;.s ~
lAY\\\ \ '\
Co'iV'QS 1-0
'* O~ c;.\\ CAM~
v::.
ff---, -=. J
.
'\ <>
O-Y\c\ -\NV\
0. S\-b-p ~
~..QY\
l-eV'-IL~'j \S
V\
~O-Q
-or \ 0'& t)
lA
0'
"tV'
2""
.
come
U ( ~)
-:: \.-
the motion of the particle aver a long period of time.
\)S'\ ~ \
v=J~o--1J:3
W'
V o..v-\-"\
e.
1-
~Q
-= V\ ~+ ~'102.
•
\ O~
V-e \ 0 CA~
bo....c.~
-ttl
~~
\ u;).
A partt~le
of mass M ~ 8 kg
moves along the x-axis under
the influence
force.
of a conservative
The corresponding
I
pote~tial energy, U(x). is
shown.
,
-~ -2.. -1
(A)
List all positions
-2.5m , x , 4mJ
of equilibrium
(stable or unstable)
L ---.lit:.?_~~~..E_,,~1_1.
__
. ui 1. Pos I n x
I
-L
o
in the region
by check marks.
and indicate the kind of stability
-l---~"-".t-,,-./e E'Iuil,.j
---------t-------------- -----------
3
"/S
(B)
Initially.
table
V_
.--1.
the
pa.rti.c.l,=:
\¥here the
particle
enerQV
100 J
150 J
be aft~r
a long
the
I
x .• + ••• x •
-
.~
-./
-./
v'
x
../
-../
x
x
entry
tranned
x
x
in the
time.
x
100 J
150 J
cht!ck m,'lr1<s (under
listed
position
right
Left
20 J
20 J
will
For ~ach case
l!l.
moving
Initial
kinetic
a:: x = -J
by atJpr~pr13t.:.
b'~lnw indicate
"position")
is
V
-