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simple harmonic motion for the
rough horizontal surface 1.50 m
mass shown is equivalent to the
long. The coefficient of kinetic
period for two springs in parallel.
friction between the block and the
rough horizontal surface is 0.45.
The block then collides with a
horizontal spring of k = 20.0 N/m.
Find the maximum compression of
the spring.
Diagram for problem 38.
*39. A mass is attached to a
Diagram for problem 45.
horizontal spring. The mass is given
an initial amplitude of 10.0 cm on a
*46. A nail is placed in the wall
rough surface and is then released
at a distance of l/2 from the top, as
to oscillate in simple harmonic
shown in the diagram. A pendulum
motion. If 10.0% of the energy is
Diagram for problem 42.
of length 85.0 cm is released from
lost per cycle due to the friction of
position 1. (a) Find the time it takes
the mass moving over the rough
*43. A 335-g disk that is free to
for the pendulum bob to reach
surface, find the maximum
rotate about its axis is connected to
position 2. When the bob of the
displacement of the mass after 1, 2,
a spring that is stretched 35.0 cm.
pendulum reaches position 2, the
4, 6, and 8 complete oscillations.
The spring constant is 10.0 N/m.
string hits the nail. (b) Find the
*40. Find the maximum
When the disk is released it rolls
time it takes for the pendulum bob
amplitude of vibration after 2
without slipping as it moves toward
to reach position 3.
periods for a 85.0-g mass executing
the equilibrium position. Find the
simple harmonic motion on a rough
speed of the disk when the
horizontal surface of µ = 0.350. The
k
displacement of the spring is equal
spring constant is 24.0 N/m and the
to -17.5 cm.
initial amplitude is 20.0 cm.
*44. A 25.0-g ball moving at a
41. A 240-g mass slides down a
velocity of 200 cm/s to the right
circular chute without friction and
makes an inelastic collision with a
collides with a horizontal spring, as
200-g block that is initially at rest
shown in the diagram. If the
on a frictionless surface. There is a
original position of the mass is 25.0
hole in the large block for the small
cm above the table top and the
ball to fit into. If k = 10 N/m, find
spring has a spring constant of 18
Diagram for problem 46.
Chapter 11 Simple Harmonic Motion 11-21
*47. A spring is attached to the where Ä is the torque acting on the mass, (g) the maximum restoring
top of an Atwood s machine as body, ¸ is the angular displacement, force F , and (h) the velocity of
Rmax
shown. The spring is stretched to A and C is a constant, like the spring the mass at the displacement x =
= 10 cm before being released. Find constant. Use Newton s second law 0.08 m. (i) Plot the displacement x,
the velocity of m when x = -A/2. for rotational motion to show velocity v, acceleration a, and the
2
Assume m = 28.0 g, m = 43.0 g, restoring force F at any time t.
1 2 R
and k = 10.0 N/m. ± = C ¸ 53. Conservation of Energy and
I the Vibrating Horizontal Spring. A
mass m = 0.350 kg is attached to a
Use the analogy between the horizontal spring. The mass is then
linear result, a = -É2x, to show that pulled a distance x = A = 0.200 m
the frequency of vibration of an from its equilibrium position and
object executing angular simple when released the mass executes
harmonic motion is given by simple harmonic motion. Find
(a) the total energy E of the mass
tot
when it is at its maximum
1 C
f =
displacement A from its equilibrium
2À I
position. When the mass is at the
displacement x = 0.120 m find,
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(b) its potential energy PE, (c) its
50. Simple Pendulum. Calculate
kinetic energy KE, and (d) its speed
the period T of a simple pendulum
v. (e) Plot the total energy, potential
located on a planet having a
energy, and kinetic energy of the
gravitational acceleration of g =
mass as a function of the
9.80 m/s2, if its length l = 1.00 m is
displacement x. The spring constant
increased from 1 to 10 m in steps of
k = 35.5 N/m.
1.00 m. Plot the results as the
Diagram for problem 47. 54. Conservation of Energy and
period T versus the length l.
the Vibrating Vertical Spring. A
51. Simple Harmonic Motion.
*48. A 280-g block is connected
mass m = 0.350 kg is attached to a
The displacement x of a body
to a spring on a rough inclined
vertical spring. The mass is at a
undergoing simple harmonic motion
plane that makes an angle of 35.50
height h = 1.50 m from the floor.
is given by the formula x = A cos Ét,
with the horizontal. The block is
The mass is then pulled down a
where A is the amplitude of the
pulled down the plane a distance A
distance A = 0.220 m from its
vibration, É is the angular
= 20.0 cm, and is then released. The
equilibrium position and when
frequency in rad/s, and t is the time
spring constant is 40.0 N/m and the
released executes simple harmonic
in seconds. Plot the displacement x
coefficient of kinetic friction is
motion. Find (a) the total energy of
as a function of t for an amplitude A
0.100. Find the speed of the block
the mass when it is at its maximum
= 0.150 m and an angular frequency
when the displacement x = -A/2.
displacement A below its
É = 5.00 rad/s as t increases from 0
equilibrium position, (b) the
to 2 s in 0.10 s increments.
gravitational potential energy when
52. The Vibrating Spring. A
it is at the displacement x = 0.120
mass m = 0.500 kg is attached to a
m, (c) the elastic potential energy
spring on a smooth horizontal table.
when it is at the same displacement
An applied force F = 4.00 N is used
A
x, (d) the kinetic energy at the
to stretch the spring a distance x =
displacement x, and (e) the speed of
0.150 m. (a) Find the spring
the mass when it is at the
constant k of the spring. The mass
displacement x. The spring constant
is returned to its equilibrium
k = 35.5 N/m.
Diagram for problem 48.
position and then stretched to a
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value A = 0.15 m and then released.
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49. The rotational analog of
The mass then executes simple
simple harmonic motion, is angular
harmonic motion. Find (b) the
simple harmonic motion, wherein a
angular frequency É, (c) the
body rotates periodically clockwise
frequency f, (d) the period T, (e) the
and then counterclockwise. Hooke s
maximum velocity v of the
max
law for rotational motion is given by
vibrating mass, (f) the maximum
Ä = -C ¸
acceleration a of the vibrating
max
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11-22 Vibratory Motion, Wave Motion and Fluids [ Pobierz całość w formacie PDF ]