Engenharia Química/Civil
Física 2
Prof. Márcio Boer Ribeiro
Guia de Estudo: MHS
•O movimento periódico é aquele que se repete em um ciclo definido. Ele ocorre quando um corpo
possui uma posição de equilíbrio estável e uma força restauradora que atua sobre o corpo quando ele
é deslocado da sua posição de equilíbrio. O período Té o tempo necessário para completar um ciclo.
A frequência fé o número de ciclos por unidade de tempo. A frequência angular ωé 2πvezes a
frequência.
461
CHAPTER 14 SUMMARY
Simple harmonic motion: If the restoring force in
periodic motion is directly proportional to the displace-
ment x, the motion is called simple harmonic motion
(SHM). In many cases this condition is satisfied if the
displacement from equilibrium is small. The angular
frequency, frequency, and period in SHM do not depend
on the amplitude, but only on the mass mand force con-
stant k. The displacement, velocity, and acceleration in
SHM are sinusoidal functions of time; the amplitude A
and phase angle of the oscillation are determined by
the initial position and velocity of the body. (See Exam-
ples 14.2, 14.3, 14.6, and 14.7.)
f
Fx
Energy in simple harmonic motion: Energy is conserved
in SHM. The total energy can be expressed in terms of
the force constant kand amplitude A. (See Examples
14.4 and 14.5.)
Angular simple harmonic motion: In angular SHM, the
frequency and angular frequency are related to the
moment of inertia Iand the torsion constant .k
(14.3)
(14.4)
(14.10)
(14.11)
(14.12)
(14.13)
x=Acos1vt+f2
T=1
ƒ=2pAm
k
ƒ=v
2p=1
2pAk
m
v=Ak
m
a
x=Fx
m=-
k
mx
Fx=-kx
(14.21)
E=1
2mvx2+1
2kx2=1
2kA2=constant
(14.24)
v=Ak
I and ƒ=1
2pAk
I
Simple pendulum: Asimple pendulum consists of a point
mass mat the end of a massless string of length L. Its
motion is approximately simple harmonic for suffi-
ciently small amplitude; the angular frequency, fre-
quency, and period then depend only on gand L, not on
the mass or amplitude. (See Example 14.8.)
(14.32)
(14.33)
(14.34)
T=2p
v=1
ƒ=2pAL
g
ƒ=v
2p=1
2pAg
L
v=Ag
L
Periodic motion: Periodic motion is motion that repeats
itself in a definite cycle. It occurs whenever a body has a
stable equilibrium position and a restoring force that
acts when it is displaced from equilibrium. Period Tis
the time for one cycle. Frequency is the number of
cycles per unit time. Angular frequency is times
the frequency. (See Example 14.1.)
2pv
ƒ
(14.1)
(14.2)
v=2pƒ=2p
T
ƒ=1
T T=1
ƒ
Fx
a
x
x
x
n
mg
y
n
mg
y
Fx
a
x
x
n
mg
y
x = 2Ax = 0
x,0x.0
x = A
x
2T
O
A
Tt
x
2A
Energy
x
E5K1U
OA2A
U
K
u
t
z
SpringBalance wheel
Spring torque tzopposes
angular displacement u.
L
T
mg sin u
mg
mg cos u
u
Physical pendulum: Aphysical pendulum is any body
suspended from an axis of rotation. The angular fre-
quency and period for small-amplitude oscillations are
independent of amplitude, but depend on the mass m,
distance dfrom the axis of rotation to the center of grav-
ity, and moment of inertia Iabout the axis. (See Exam-
ples 14.9 and 14.10.)
(14.38)
(14.39)T=2pAI
mgd
v=
B
mgd
Id
z
mg sinu
mg
mg cosu
cg
O
dsin uu
f=1
T(1)
ω=2πf=2π
T(2)
•O movimento harmônico simples: quando a força resultante for uma força restauradora Fxdire-
tamente proporcional ao deslocamento x, o movimento denomina-se movimento harmônico simples
(MHS). Em muitos casos, essa condição é satisfeita se o deslocamento a partir do equilíbrio for pe-
queno. A frequência angular, a frequência e o período em MHS não dependem da amplitude, apenas
da massa me da constante da mola k. O deslocamento, a velocidade e a aceleração em MHS são
funções senoidais do tempo; a amplitude Ae o ângulo de fase φda oscilação são determinados pela
posição inicial e velocidade do corpo.
461
CHAPTER 14 SUMMARY
Simple harmonic motion: If the restoring force in
periodic motion is directly proportional to the displace-
ment x, the motion is called simple harmonic motion
(SHM). In many cases this condition is satisfied if the
displacement from equilibrium is small. The angular
frequency, frequency, and period in SHM do not depend
on the amplitude, but only on the mass mand force con-
stant k. The displacement, velocity, and acceleration in
SHM are sinusoidal functions of time; the amplitude A
and phase angle of the oscillation are determined by
the initial position and velocity of the body. (See Exam-
ples 14.2, 14.3, 14.6, and 14.7.)
f
Fx
Energy in simple harmonic motion: Energy is conserved
in SHM. The total energy can be expressed in terms of
the force constant kand amplitude A. (See Examples
14.4 and 14.5.)
Angular simple harmonic motion: In angular SHM, the
frequency and angular frequency are related to the
moment of inertia Iand the torsion constant .k
(14.3)
(14.4)
(14.10)
(14.11)
(14.12)
(14.13)
x=Acos1vt+f2
T=1
ƒ=2pAm
k
ƒ=v
2p=1
2pAk
m
v=Ak
m
a
x=Fx
m=-
k
mx
Fx=-kx
(14.21)
E=1
2mvx2+1
2kx2=1
2kA2=constant
(14.24)
v=Ak
I and ƒ=1
2pAk
I
Simple pendulum: Asimple pendulum consists of a point
mass mat the end of a massless string of length L. Its
motion is approximately simple harmonic for suffi-
ciently small amplitude; the angular frequency, fre-
quency, and period then depend only on gand L, not on
the mass or amplitude. (See Example 14.8.)
(14.32)
(14.33)
(14.34)
T=2p
v=1
ƒ=2pAL
g
ƒ=v
2p=1
2pAg
L
v=Ag
L
Periodic motion: Periodic motion is motion that repeats
itself in a definite cycle. It occurs whenever a body has a
stable equilibrium position and a restoring force that
acts when it is displaced from equilibrium. Period Tis
the time for one cycle. Frequency is the number of
cycles per unit time. Angular frequency is times
the frequency. (See Example 14.1.)
2pv
ƒ
(14.1)
(14.2)
v=2pƒ=2p
T
ƒ=1
T T=1
ƒ
Fx
a
x
x
x
n
mg
y
n
mg
y
Fx
a
x
x
n
mg
y
x = 2Ax = 0
x,0x.0
x = A
x
2T
O
A
Tt
x
2A
Energy
x
E5K1U
OA2A
U
K
u
t
z
SpringBalance wheel
Spring torque tzopposes
angular displacement u.
L
T
mg sin u
mg
mg cos u
u
Physical pendulum: Aphysical pendulum is any body
suspended from an axis of rotation. The angular fre-
quency and period for small-amplitude oscillations are
independent of amplitude, but depend on the mass m,
distance dfrom the axis of rotation to the center of grav-
ity, and moment of inertia Iabout the axis. (See Exam-
ples 14.9 and 14.10.)
(14.38)
(14.39)T=2pAI
mgd
v=
B
mgd
Id
z
mg sinu
mg
mg cosu
cg
O
dsin uu
F=−kx (3)
ax=−
k
mx=−ω2x(4)
ω=rk
m(5)
f=ω
2π=1
2πrk
m(6)
T=1
f=2πrm
k(7)
x=Acos(ωt+φ)(8)
1
