Download Simple harmonic Motion Answers

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Simple Harmonic Motion Unit
Written Problems
1.
An object af maSS0.25 kg is acted on by an elastic restaring force of farce constant k=.?W/m.
(a) Construct a graph af elastic potential energy U as a function of displacement, x, over a range af
from
.0.3 m ta +0.3 m, calculating U(xY for every 0.05
m.
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The object is set into oscillation with <Ininitial potential energy of 0.6 J and an initial kinetic energy af 0.2 J.
Answer the fallowing questions by reference to the graph.
D.g J
(b) What is the amplitude of oscillation?
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(e)
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What is the potential energy when the displucernent is one-half the amplitude?
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(d)
At what displacement are the kinetic and potential energies equal?
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(e)
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What is the s~ed of the object at the midpoint of its path (that is, at x=O)?
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Find
(f)
the period, T,
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the frequency, f and
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(h) the angular frequency,
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2.
A certain simple pendulum has a period on Earth of 2.0 s. What is its period on the surface of the moon
where g = 1.7 m/ S2?
3.
A block of mass ml attached to 0 horizontal spring of force constant k is moving with simple harmonic
motion of amplitude A. At the instant it passes through its equilibrium position, a lump of putty of mass m2 is
dropped vertically onto the block from a very small height and sticks to it.
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(b) Was there a loss of mechanical~YT!t
so, where did it go?
between the final and the initial mechanical energy.
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e ratio
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(c) Would the answers be the same if the putty had been dropped on the block when it
was at one end of its path? Explain (with calculations or words).
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4.
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A solid disk/of radius R = 12 em oscillates as a physical pendulum abaut an axis perpendicular to the plane
of the disk at a distance r from its center. Calculate the period of oscillation when (a) r = R/2 and
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5.
Two springs with the same unstretched length but different force constants, k, and kz are attached to
a block of mass m on a level, frictionless surface. Calculate the effective farce constant in each of the three
cases (a), (b) and (c).
(b)
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(c)
K
2008M3.
In an experiment to detennine the spring constant of an elastic cord oflength 0.60 m, a student hangs the
cord from a rod as represented above and then attaches a variety of weights to the cord. For each weight, the student
allows the weight to hang in equilibrium and then measures the entire length of the cord. The data are recorded in the
table below:
o
10
15
0.60
0.97
1.24
20
25
1.37 1.64
(a) Use the data to plot a graph of weight versus length on the axes below. Sketch a best-fit straight line through the
data.
30
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(b) Use the best-fit line you sketched in part (a) to detennine an experimental value for the spring constant kofthe
cord.
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:[he student now attaches an object of unknown mass m to the cord and holds the object adjacent to the
'point at which the top of the cord is tied to the rod, as represented above. When the object is released
from rest, it falls 1.5 m before stopping and turning around. Assume that air resistance is negligible.
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0.'0<;
0
(c) Calculate
the value of the unknown mass m of the object.
= '/2 k)(2
~h
M~
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:: 0, u::;lie
'0
2(U')( ,.'.))
(d) i. Calculate
aA
how far down the object has fallen at the moment it attains its maximum speed._
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Explain why this is the poii?t a'i'which the object has its maximum speed.
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Calculate
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the maximum
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C
r-~ 0
speed of the object.
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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.
X2-
x(m)
F(N)
0.05
4
0.10
17
,01
0.15
38
6225
0.20
68
O,D
0.25
106
OdX)2S,
, 0 fo"LS
Assume that the magnitude of the force applied by the spring is of the form F(x) = A.!.
a. Which quantities should be graphed in order to yield a stmight line whose slope could be used to calculate a numerical
value for A?
~ vs X "1..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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d. Using your graph, calculate A .
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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. Calculatey the work done by the cart in compressing the spring 0.10 m from its equilibrium length.
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f. Calculate the speed of the cart just before it strikes the spring.
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2004M3(revised).
A uniform rod of mass M and length L is attached to a pivot of negligible friction as shown above. The pivot.,
located at a distance L/3 from the lell end of the rod. Express all answers in terms of the given quantities and fundamental
constants.
~
a.
Calculate the rotational inertia of the rod about the pivot.
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q
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b.
Tbe rod is then released from reZt~~; the honzontal position shown above. Calculate the linear speed of the bottom_end of
the rod when the rod passes through the vertical.
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1:..L/2
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c.
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The rod is brought to rest in the vertical position shown above and hangs freely. It is then displaced slightly from this
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d.
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Calculate the period of oscillation as it swings, assuming that the angle of osci Iation is small.
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2003M2. An ideal spring is hung from the ceiling and a pan of mass M is suspended from the end of the spring, stretching it
a distance D as shown above. A piece of clay, also of mass M, is then dropped from a height H onto the pan and sticks to it.
Express all algebraic answers in terms of the given quantities and fundamental constants.
a. Determine the speed of the clay at the instant it hits the pan.
~~H--='
V ~
b.
v,t\V
'( ~
1-
1;';)5 l-t
Determine the speed of the pan just after the clay strikes it.
~1D_
Determine the period of the simple harmonic motion that ensues.
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d.
v;l\5
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Determine the distance the spring is stretched from);i
maximum. Justify your answer.
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initial unstretched length) at the moment the speed of the pan is a
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0
-10-::;
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e.
The clay is now removed from the pan and the pan is returned to equilibrium at the end of the spring. A rubber ball, also of
mass M. is dropped from the same height H onto the pan, and aller the collision is caught in midair before hitting anything
else.
Indicate below whether the period of the resulting simple harmonic motion of the pan is greater than, less than, or the same as it
was in pan (c).
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__
Greater than
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r °essthan
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The same as
Justify your answer.
T
B
0.45 m
1989M3. A 2-kilogram block is dropped from a height of 0.45 meter ahove an uncompressed spring, as shoWn above. The spring
has an elastic constant of 200 newtons per meter and negligible mass. The block strikes the end of the spring and sticks to it.
a. Determine the speelof the block at the instant it hits the end of the spring.
~ h~ Yj..f>Ilv
1-
V -TJ.Lq.~J(''fl;)
b.
Determine the period of the simple harmonic mOJionJhat ensues.
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c.
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Q.:: 0
K)(
m9
d.
Determine the maximum compression of the spring.
t-VY\5V\. ::: \/v iL-X 2y):z)(o?,q~)1- f- ;;'{Q/6)X-C:
'/(f-l\V
e.
D
Determine the distance that the spring is compressed at the instant the speed of the block is maximum.
L
Determine the amplitude of the simple harmonic motion.
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1980MI. A small mass m, rests on but is not attached to a large mass M, that slides on its base without friction.The maximum frictional force between m, and M, is f. A spring of spring constant k is attached to the
large mass M, and to the wall as shown above.
a. Determine the maximum horizontal acceleration that M2 may have without causing ml to sJip.
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::::0.'2.
-==- (Y) \ 0..
J
('--
Ct ~ ::r
-m,
b. Determine the maximum amp 't de Alor simple harmonic motion of the twn masses
slip on M,.
if they
are to move together, i.e., m, must not
c. The two-mass combination is pulled to the right the maximum amplitude A found in part (b) and released. Describe the frictional
force on th:-small mass m, during the first half cycle ofoscil/ation.
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utd-J.
d.
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af
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The two-mass combination is now pulled to the right a distance of A' greater than A and released.
i. Determine the acceleration of ml at the instant the masses are released.
-
J;;(Y1J),
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L
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Determine the acceleration ofM, at the instant the masses are relea~ed.
k A\ - f
::M1- CA..
.
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