Physics For Scientists And Engineers 6E - part 120

4. (a) A hanging spring stretches by 35.0 cm when an object

of mass 450 g is hung on it at rest. In this situation, we de-
fine its position as x ! 0. The object is pulled down an ad-
ditional 18.0 cm and released from rest to oscillate without
friction. What is its position x at a time 84.4 s later?
(b)

What If? A hanging spring stretches by 35.5 cm when

an object of mass 440 g is hung on it at rest. We define this
new position as x ! 0. This object is also pulled down an
additional 18.0 cm and released from rest to oscillate with-
out friction. Find its position 84.4 s later. (c) Why are the
answers to (a) and (b) different by such a large percentage
when the data are so similar? Does this circumstance reveal
a fundamental difficulty in calculating the future? (d) Find
the distance traveled by the vibrating object in part (a).
(e) Find the distance traveled by the object in part (b).

A particle moving along the x axis in simple har-

monic motion starts from its equilibrium position, the ori-
gin, at t ! 0 and moves to the right. The amplitude of its
motion is 2.00 cm, and the frequency is 1.50 Hz. (a) Show
that the position of the particle is given by

Determine (b) the maximum speed and the earliest time
(t # 0) at which the particle has this speed, (c) the maxi-
mum acceleration and the earliest time (t # 0) at which
the particle has this acceleration, and (d) the total dis-
tance traveled between t ! 0 and t ! 1.00 s.

The initial position, velocity, and acceleration of an object
moving in simple harmonic motion are x

, v

, and a

; the

angular frequency of oscillation is '. (a) Show that the
position and velocity of the object for all time can be
written as

(b) If the amplitude of the motion is A, show that

7. A simple harmonic oscillator takes 12.0 s to undergo five

complete vibrations. Find (a) the period of its motion,
(b) the frequency in hertz, and (c) the angular frequency
in radians per second.

8. A vibration sensor, used in testing a washing machine, con-

sists of a cube of aluminum 1.50 cm on edge mounted on
one  end  of  a  strip  of  spring  steel  (like  a  hacksaw  blade)
that lies in a vertical plane. The mass of the strip is small
compared to that of the cube, but the length of the strip is
large compared to the size of the cube. The other end of
the strip is clamped to the frame of the washing machine,
which  is  not  operating.  A  horizontal  force  of  1.43 N  ap-
plied to the cube is required to hold it 2.75 cm away from
its equilibrium position. If the cube is released, what is its
frequency of vibration?
A 7.00-kg object is hung from the bottom end of a vertical
spring fastened to an overhead beam. The object is set into
vertical  oscillations  having  a  period  of  2.60 s.  Find  the
force constant of the spring.

ax ! v

v(t) ! "x

sin 't % v

cos 't

x(t) ! x

cos 't %

sin 't

x ! (2.00 cm)sin(3.00)t)

Problems

477

10. A piston in a gasoline engine is in simple harmonic

motion. If the extremes of its position relative to its center
point  are  & 5.00 cm,  find  the  maximum  velocity  and
acceleration  of  the  piston  when  the  engine  is  running  at
the rate of 3 600 rev/min.
A 0.500-kg object attached to a spring with a force constant
of  8.00 N/m  vibrates  in  simple  harmonic  motion  with  an
amplitude of 10.0 cm. Calculate (a) the maximum value of
its speed and acceleration, (b) the speed and acceleration
when the object is 6.00 cm from the equilibrium position,
and (c) the time interval required for the object to move
from x ! 0 to x ! 8.00 cm.

12.

A 1.00-kg glider attached to a spring with a force constant
of  25.0 N/m  oscillates  on  a  horizontal,  frictionless  air
track.  At  t ! 0  the  glider  is  released  from  rest  at
x ! " 3.00 cm.  (That  is,  the  spring  is  compressed  by
3.00 cm.) Find (a) the period of its motion, (b) the maxi-
mum values of its speed and acceleration, and (c) the posi-
tion, velocity, and acceleration as functions of time.

13. A 1.00-kg object is attached to a horizontal spring. The

spring is initially stretched by 0.100 m, and the object is re-
leased from rest there. It proceeds to move without fric-
tion. The next time the speed of the object is zero is
0.500 s later. What is the maximum speed of the object?

14.

A particle that hangs from a spring oscillates with an angu-
lar frequency '. The spring is suspended from the ceiling
of an elevator car and hangs motionless (relative to the
elevator car) as the car descends at a constant speed v. The
car then stops suddenly. (a) With what amplitude does the
particle oscillate? (b) What is the equation of motion for
the particle? (Choose the upward direction to be positive.)

Section 15.3 Energy of the Simple Harmonic

Oscillator

15. A block of unknown mass is attached to a spring with a

spring  constant  of  6.50 N/m  and  undergoes  simple  har-
monic  motion  with  an  amplitude  of  10.0 cm.  When  the
block  is  halfway  between  its  equilibrium  position  and  the
end point, its speed is measured to be 30.0 cm/s. Calculate
(a)  the  mass  of  the  block,  (b)  the  period  of  the  motion,
and (c) the maximum acceleration of the block.

16. A 200-g block is attached to a horizontal spring and executes

simple harmonic motion with a period of 0.250 s. If the total
energy of the system is 2.00 J, find (a) the force constant of
the spring and (b) the amplitude of the motion.

An automobile having a mass of 1 000 kg is driven

into a brick wall in a safety test. The bumper behaves like a
spring of force constant 5.00 , 10

N/m and compresses

3.16 cm as the car is brought to rest. What was the speed
of the car before impact, assuming that no mechanical
energy is lost during impact with the wall?

18. A block–spring system oscillates with an amplitude of

3.50 cm. If the spring constant is 250 N/m and the mass of
the block is 0.500 kg, determine (a) the mechanical en-
ergy of the system, (b) the maximum speed of the block,
and (c) the maximum acceleration.

19.

A 50.0-g object connected to a spring with a force constant
of 35.0 N/m oscillates on a horizontal, frictionless surface

17.

11.

with an amplitude of 4.00 cm. Find (a) the total energy of
the system and (b) the speed of the object when the posi-
tion is 1.00 cm. Find (c) the kinetic energy and (d) the po-
tential energy when the position is 3.00 cm.

20.

A  2.00-kg  object  is  attached  to  a  spring  and  placed  on  a
horizontal, smooth surface. A horizontal force of 20.0 N is
required  to  hold  the  object  at  rest  when  it  is  pulled
0.200 m from its equilibrium position (the origin of the x
axis). The object is now released from rest with an initial
position  of  x

0.200 m, and it subsequently undergoes

simple  harmonic  oscillations.  Find  (a)  the  force  constant
of  the  spring,  (b)  the  frequency  of  the  oscillations,  and
(c) the  maximum  speed  of  the  object.  Where  does  this
maximum  speed  occur?  (d)  Find  the  maximum  accelera-
tion of the object. Where does it occur? (e) Find the total
energy  of  the  oscillating  system.  Find  (f)  the  speed  and
(g) the acceleration of the object when its position is equal
to one third of the maximum value.

21. The amplitude of a system moving in simple harmonic mo-

tion is doubled. Determine the change in (a) the total
energy, (b) the maximum speed, (c) the maximum accel-
eration, and (d) the period.

22.

A  65.0-kg  bungee  jumper  steps  off  a  bridge  with  a  light
bungee  cord  tied  to  herself  and  to  the  bridge  (Figure
P15.22). The unstretched length of the cord is 11.0 m. She
reaches the bottom of her motion 36.0 m below the bridge
before bouncing back. Her motion can be separated into
an  11.0-m  free  fall  and  a  25.0-m  section  of  simple  har-
monic oscillation. (a) For what time interval is she in free
fall?  (b)  Use  the  principle  of  conservation  of  energy  to
find  the  spring  constant  of  the  bungee  cord.  (c)  What  is
the  location  of  the  equilibrium  point  where  the  spring
force  balances  the  gravitational  force  acting  on  the
jumper? Note that this point is taken as the origin in our
mathematical  description  of  simple  harmonic  oscillation.
(d)  What  is  the  angular  frequency  of  the  oscillation?
(e) What time interval is required for the cord to stretch
by 25.0 m? (f ) What is the total time interval for the entire
36.0-m drop?

478

C H A P T E R 15 • Oscillatory Motion

A particle executes simple harmonic motion with an am-
plitude of 3.00 cm. At what position does its speed equal
half its maximum speed?

24.

A cart attached to a spring with constant 3.24 N/m vibrates
with  position  given  by  x ! (5.00 cm) cos(3.60t rad/s).
(a)  During  the  first  cycle,  for  0 $ t $ 1.75 s,  just  when  is
the  system’s  potential  energy  changing  most  rapidly  into
kinetic  energy?  (b)  What  is  the  maximum  rate  of  energy
transformation?

Section 15.4 Comparing Simple Harmonic Motion

with Uniform Circular Motion

25. While riding behind a car traveling at 3.00 m/s, you notice

that one of the car’s tires has a small hemispherical bump
on its rim, as in Figure P15.25. (a) Explain why the bump,
from your viewpoint behind the car, executes simple har-
monic motion. (b) If the radii of the car’s tires are
0.300 m, what is the bump’s period of oscillation?

23.

26.

Consider  the  simplified  single-piston  engine  in  Figure
P15.26.  If  the  wheel  rotates  with  constant  angular  speed,
explain  why  the  piston  rod  oscillates  in  simple  harmonic
motion.

Section 15.5 The Pendulum

Figure P15.22 Problems 22 and 58.

Telegraph Colour Library / FPG International

Bump

Figure P15.25

Piston

x = "A

x(t )

Figure P15.26

Problem 60 in Chapter 1 can also be assigned with this
section.

27. A man enters a tall tower, needing to know its height. He

notes that a long pendulum extends from the ceiling
almost to the floor and that its period is 12.0 s. (a) How
tall is the tower? (b)

What If? If this pendulum is taken to

the Moon, where the free-fall acceleration is 1.67 m/s

what is its period there?

28. A “seconds pendulum” is one that moves through its

equilibrium position once each second. (The period of
the pendulum is precisely 2 s.) The length of a seconds
pendulum is 0.992 7 m at Tokyo, Japan and 0.994 2 m at
Cambridge, England. What is the ratio of the free-fall ac-
celerations at these two locations?

29. A rigid steel frame above a street intersection supports

standard traffic lights, each of which is hinged to hang im-
mediately below the frame. A gust of wind sets a light
swinging in a vertical plane. Find the order of magnitude
of its period. State the quantities you take as data and their
values.

30. The angular position of a pendulum is represented by the

equation . ! (0.320 rad)cos 't, where . is in radians and
' !

4.43 rad/s. Determine the period and length of the

pendulum.

A simple pendulum has a mass of 0.250 kg and a

length of 1.00 m. It is displaced through an angle of 15.0°
and then released. What are (a) the maximum speed,
(b) the maximum angular acceleration, and (c) the maxi-
mum restoring force?

What If? Solve this problem by us-

ing the simple harmonic motion model for the motion of
the pendulum, and then solve the problem more precisely
by using more general principles.

32.

Review problem. A simple pendulum is 5.00 m long.
(a) What is the period of small oscillations for this pendu-
lum if it is located in an elevator accelerating upward at
5.00 m/s

? (b) What is its period if the elevator is acceler-

ating downward at 5.00 m/s

? (c) What is the period of

this pendulum if it is placed in a truck that is accelerating
horizontally at 5.00 m/s

A particle of mass m slides without friction inside a hemi-
spherical bowl of radius R. Show that, if it starts from rest
with  a  small  displacement  from  equilibrium,  the  particle
moves  in  simple  harmonic  motion  with  an  angular  fre-
quency  equal  to  that  of  a  simple  pendulum  of  length  R.
That is,

34.

A small object is attached to the end of a string to

form a simple pendulum. The period of its harmonic mo-
tion  is  measured  for  small  angular  displacements  and
three lengths, each time clocking the motion with a stop-
watch for 50 oscillations. For lengths of 1.000 m, 0.750 m,
and  0.500 m,  total  times  of  99.8 s,  86.6 s,  and  71.1 s  are
measured for 50 oscillations. (a) Determine the period of
motion for each length. (b) Determine the mean value of
g obtained  from  these  three  independent  measurements,
and compare it with the accepted value. (c) Plot T

versus

L, and obtain a value for g from the slope of your best-fit
straight-line graph. Compare this value with that obtained
in part (b).

' !

√

g/R

33.

31.

Problems

479

A physical pendulum in the form of a planar body moves
in simple harmonic motion with a frequency of 0.450 Hz.
If the pendulum has a mass of 2.20 kg and the pivot is
located 0.350 m from the center of mass, determine the
moment of inertia of the pendulum about the pivot point.

36.

A  very  light  rigid  rod  with  a  length  of  0.500 m  extends
straight  out  from  one  end  of  a  meter  stick.  The  stick  is
suspended from a pivot at the far end of the rod and is set
into  oscillation.  (a)  Determine  the  period  of  oscillation.
Suggestion: Use the parallel-axis theorem from Section 10.5.
(b)  By  what  percentage  does  the  period  differ  from  the
period of a simple pendulum 1.00 m long?

37.

Consider the physical pendulum of Figure 15.18. (a) If its
moment of inertia about an axis passing through its center
of mass and parallel to the axis passing through its pivot
point is I

, show that its period is

where d is the distance between the pivot point and center
of mass. (b) Show that the period has a minimum value
when d satisfies md

38. A torsional pendulum is formed by taking a meter stick of

mass  2.00 kg,  and  attaching  to  its  center  a  wire.  With  its
upper end clamped, the vertical wire supports the stick as
the  stick  turns  in  a  horizontal  plane.  If  the  resulting
period  is  3.00 minutes,  what  is  the  torsion  constant  for
the wire?

39.

A clock balance wheel (Fig. P15.39) has a period of oscilla-
tion of 0.250 s. The wheel is constructed so that its mass of
20.0 g is concentrated around a rim of radius 0.500 cm.
What are (a) the wheel’s moment of inertia and (b) the
torsion constant of the attached spring?

T ! 2)

√

mgd

35.

Figure P15.39

George Semple

Section 15.6 Damped Oscillations

40.

Show that the time rate of change of mechanical energy
for a damped, undriven oscillator is given by dE/dt ! " bv

and hence is always negative. Proceed as follows: Differen-
tiate the expression for the mechanical energy of an oscil-
lator,

, and use Equation 15.31.

41. A pendulum with a length of 1.00 m is released from an

initial angle of 15.0°. After 1 000 s, its amplitude has been
reduced by friction to 5.50°. What is the value of b/2m?

42.

Show that Equation 15.32 is a solution of Equation 15.31
provided that b

4mk.

43.

A 10.6-kg object oscillates at the end of a vertical spring
that has a spring constant of 2.05 , 10

N/m. The effect

of air resistance is represented by the damping coefficient
b ! 3.00 N 4 s/m. (a) Calculate the frequency of the
damped oscillation. (b) By what percentage does the am-
plitude of the oscillation decrease in each cycle? (c) Find
the time interval that elapses while the energy of the sys-
tem drops to 5.00% of its initial value.

Section 15.7 Forced Oscillations

44. The front of her sleeper wet from teething, a baby rejoices

in the day by crowing and bouncing up and down in her
crib.  Her  mass  is  12.5 kg,  and  the  crib  mattress  can  be
modeled as a light spring with force constant 4.30 kN/m.
(a) The baby soon learns to bounce with maximum ampli-
tude  and  minimum  effort  by  bending  her  knees  at  what
frequency? (b) She learns to use the mattress as a trampo-
line—losing  contact  with  it  for  part  of  each  cycle—when
her amplitude exceeds what value?

45. A 2.00-kg object attached to a spring moves without

friction  and  is  driven  by  an  external  force  given  by  F !
(3.00 N)sin(2)t).  If  the  force  constant  of  the  spring  is
20.0 N/m,  determine  (a)  the  period  and  (b)  the  ampli-
tude of the motion.

46.

Considering an undamped, forced oscillator (b ! 0), show
that Equation 15.35 is a solution of Equation 15.34, with
an amplitude given by Equation 15.36.

47. A weight of 40.0 N is suspended from a spring that has a

force constant of 200 N/m. The system is undamped and is
subjected to a harmonic driving force of frequency 10.0 Hz,
resulting in a forced-motion amplitude of 2.00 cm.
Determine the maximum value of the driving force.

48.

Damping is negligible for a 0.150-kg object hanging from a
light 6.30-N/m spring. A sinusoidal force with an ampli-
tude of 1.70 N drives the system. At what frequency will the
force make the object vibrate with an amplitude of
0.440 m?

49. You are a research biologist. You take your emergency

pager  along  to  a  fine  restaurant.  You  switch  the  small
pager  to  vibrate  instead  of  beep,  and  you  put  it  into  a
side  pocket  of  your  suit  coat.  The  arm  of  your  chair
presses the light cloth against your body at one spot. Fab-
ric with a length of 8.21 cm hangs freely below that spot,
with the pager at the bottom. A coworker urgently needs

E !

480

C H A P T E R 15 • Oscillatory Motion

instructions and calls you from your laboratory. The mo-
tion  of  the  pager  makes  the  hanging  part  of  your  coat
swing  back  and  forth  with  remarkably  large  amplitude.
The waiter and nearby diners notice immediately and fall
silent.  Your  daughter  pipes  up  and  says,  “Daddy,  look!
Your cockroaches must have gotten out again!” Find the
frequency at which your pager vibrates.

50.

Four people, each with a mass of 72.4 kg, are in a car with
a mass of 1 130 kg. An earthquake strikes. The driver man-
ages to pull off the road and stop, as the vertical oscilla-
tions of the ground surface make the car bounce up and
down on its suspension springs. When the frequency of the
shaking is 1.80 Hz, the car exhibits a maximum amplitude
of vibration. The earthquake ends, and the four people
leave the car as fast as they can. By what distance does the
car’s undamaged suspension lift the car body as the people
get out?

Additional Problems

A small ball of mass M is attached to the end of a uniform
rod of equal mass M and length L that is pivoted at the top
(Fig. P15.51). (a) Determine the tensions in the rod at the
pivot  and  at  the  point  P when  the  system  is  stationary.
(b) Calculate  the  period  of  oscillation  for  small  displace-
ments  from  equilibrium,  and  determine  this  period  for
L ! 2.00 m.  (Suggestions:  Model  the  object  at  the  end  of
the rod as a particle and use Eq. 15.28.)

51.

Pivot

y = 0

Figure P15.51

52.

An object of mass m

9.00 kg is in equilibrium while

connected to a light spring of constant k ! 100 N/m that
is fastened to a wall as shown in Figure P15.52a. A second
object, m

7.00 kg, is slowly pushed up against m

, com-

pressing the spring by the amount A ! 0.200 m, (see Fig-
ure P15.52b). The system is then released, and both ob-
jects start moving to the right on the frictionless surface.
(a) When m

reaches the equilibrium point, m

loses con-

tact with m

(see Fig. P15.5c) and moves to the right with

speed v. Determine the value of v. (b) How far apart are
the  objects  when  the  spring  is  fully  stretched  for  the  first
time  (D in  Fig.  P15.52d)?  (Suggestion: First  determine  the
period  of  oscillation  and  the  amplitude  of  the  m

–spring

system after m

loses contact with m

Content .. 118 119 120 121 ..