# L1 (SHM)

```Lesson One
Oscillatory
Motion
Physics
2
Mr. Ahmed El Bagoury
– A branch of science which
concerned mainly with matter in
relation to energy.
– Physics concerns with physical
and universal phenomena and
studying these phenomena using
rules.
– The rules consist of physical
quantities.
3
Physical
Quantities
Scalar
Quantity
Vector
Quantity
Mr. Ahmed El Bagoury
Scalar Quantity
4
– t is a physical quantity that is defined by its
magnitude only.
– Such as:
Mass.
Length.
Time.
Energy.
Mr. Ahmed El Bagoury
Vector Quantity
5
– It is a physical quantity that is defined by its
magnitude and direction.
– Such as
Displacement.
Force.
Velocity.
Mr. Ahmed El Bagoury
Kinds of motion
Transolatory Motion
• Motion from one position to another point
Periodic Motion
• motion of an object that regularly returns
to a given position after fixed time interval
7
Hook’s law
– The extension is directly proportional to the stretching force
𝑭𝒐𝒓𝒄𝒆 𝜶 𝑬𝒙𝒕𝒆𝒏𝒔𝒊𝒐𝒏
𝑭𝒐𝒓𝒄𝒆 = 𝑺𝒑𝒓𝒊𝒏𝒈 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕 &times;𝑬𝒙𝒕𝒆𝒏𝒔𝒊𝒐𝒏
𝑭 = 𝑲 .𝑿
𝑭
𝑲=
𝑿
𝑺𝑰 𝒖𝒏𝒊𝒕: 𝑵/𝒎
A. Bagoury
8
Spring Constant
𝑭
𝑲=
𝑿
– it is the force per unit extension.
– it shows how the spring resists being extended.
– It measures the stiffness of the spring
– The spring constant varies from one spring to another due to
the difference in:
[1] Type of the material.
[3]Thickness of spring.
𝑮𝒓𝒂𝒅𝒊𝒆𝒏𝒕 = 𝑺𝒑𝒓𝒊𝒏𝒈 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕
∆𝑭
=
∆𝑿
A. Bagoury
[4] Length of the spring.
[5]Thickness of wire from which the spring is made.
[6] Number of turns of the spring
Series Springs:
– The total extension of the system is the sum of each individual
extension of each spring
– For each spring
𝑿𝑺𝒚𝒔𝒕𝒆𝒎 = 𝒙𝟏 + 𝒙𝟐 + ⋯
𝐹
𝐹
𝐹
= + +⋯
𝑘HIJKLM 𝑘N 𝑘O
𝟏
𝒌𝒔𝒚𝒔𝒕𝒆𝒎
𝟏
𝟏
=
+
+⋯
𝒌𝟏 𝒌𝟐
– If the system is only two spring:
𝒌𝑬𝒇𝒇
𝒌𝟏 𝒌𝟐
=
𝒌𝟏 + 𝒌𝟐
Parallel Springs
– The load attached to the system is divided to each spring.
– The total extension of the system is the same as the the extension
of each.
– For each spring:
𝐹 = 𝐹N + 𝐹O + 𝐹R + ⋯
𝑘HIJKLM . 𝑥 = 𝑘N . 𝑥 + 𝑘O . 𝑥 + 𝑘R . 𝑥
𝑘TUU = 𝑘N + 𝑘O + 𝑘R
PROBLEM A spring is hung vertically (Fig. 13.2a), and an object of mass
ves
m attached to the lower end is then slowly lowered a distance d to the equilibrium point (Fig. 13.2b). (a) Find the value of the spring constant if the
spring is displaced by 2.0 cm and the mass is 0.55 kg. (b) If a second identical spring is attached to the object in parallel with the first spring (Fig.
13.2d), where is the new equilibrium point of the system? (c) What is the
effective spring constant of the two springs acting as one?
ertical Spring
S TR
This example is an application of Newton’s second law. The
r with the force
ofATEGY
gravity and
The
cau
of th
a
spring force is upward, balancing the downward force of gravity mg when
the system is in equilibrium. (See Fig. 13.2c.) Because the suspended Figure
g. 13.2a), and object
an object
is of
in mass
equilibrium, the forces on the SFobject
sum to zero, and it’s pos- the spri
s
owered a distance
to the
equibalance
sibled to
solve
for the spring constant
d k. Part (b) is solved the same way, but
ue of the spring
if the forces balancing the force of gravity. The spring constants rium, it
hasconstant
two spring
is 0.55 kg. (b)are
If a known,
second idenso the second law for equilibrium can be solved for the dis- two spri
rallel with theplacement
first spring of
(Fig.
the spring. Part (c) involves using the displacement found in part (b).
of the system?
(c)spring,
What is the
the second
The elongation d is
S
lent
mg effective spring constant of the two-spr
acting as one?
n of Newton’s second law. The
ION
nward force ofSOLUT
gravity mg
when
caused by the weight mg
of the attached object.
a
b
c
d
– (a)
𝐹 = 𝑘. 𝑥
𝑚. 𝑔 = 𝑘. 𝑥
0.55&times;9.81 = 𝑘&times;2
𝑘 = 2.7 N/cm
- (b)
Since the springs in parallel
𝑘JIJKLM = 𝑘N + 𝑘O = 2.7&times;2 = 5.4 𝑁/𝑐𝑚
𝐹 = 𝑘JIJKLM . 𝑥
0.55&times;9.81 = 5.4 x
X = 1 cm
- (c)
Since the springs in parallel
𝑘JIJKLM = 𝑘N + 𝑘O = 2.7&times;2 = 5.4 𝑁/𝑐𝑚
Elastic Potential Energy [Strain Energy]
– It is the energy stored in an elastic material due to a force that cause an
extension, compression, deformation or change in size of a body.
𝑊𝑟𝑜𝑘 = 𝑃𝐸LlmJKno
1
𝑃𝐸LlmJKno = 𝐹. ∆𝑥
2
𝐹 = 𝑘. ∆𝑥
𝟏
𝑷𝑬𝒆𝒍𝒂𝒔𝒕𝒊𝒄 = 𝒌. ∆𝒙𝟐
𝟐
– The work done in stretching or compressing a
material is always equal to the area under the
graph of force against extension.
Simple Harmonic Motion
– It is the motion of a oscillating object in which the acceleration (force)
is directly proportional to the displacement and opposite to it.
𝑎 ∝ −𝑥
Where a is the acceleration of the oscillating body.
𝒙 is the displacement covered by the object
Motion of an Object Attached to a Spring
– Equilibrium position of the system,
which we identify as x = 0
– During the oscillation of the mass m, the
spring exerts a force F called restoring
force.
Measuring Oscillation
Period T:
– It is the time taken by the vibrating body to make one complete oscillation
time
T=
No. of oscillations
Frequency f:
– It is the number of oscillations per unit time.
No. of oscillations
f=
time
1
f=
T
Amplitude
𝐱𝐨
– It is the maximum displacement away from the equilibrium position.
Phase
– The point that an oscillating mass has reached within the complete cycle of an
oscillation at certain instant with certain speed.
– It is often important to describe the phase difference between two oscillations.
630
𝑝ℎ𝑎𝑠𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 𝑝𝑎𝑡ℎ 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 &times;
𝜆
Restoring Force
– It is the force that tends to bring the system back to the equilibrium position.
𝐹J = −𝑘. 𝑥
From Newton’s 2nd Law
𝑭𝒙 = 𝒎. 𝒂𝒙
Because the oscillating mass is vibrating the x dimension only, so the resultant force
acting on it only in dimension x.
𝑚. 𝑎• − 𝑘. 𝑥
𝒌
𝒂𝒙 = − . 𝒙
𝒎
𝒌
𝒂𝒙 = − . 𝒙
𝒎
When
x= +A
When
x= 0
When
x= -A
𝒌
𝑎 = − .𝒙
𝒎
Speed=0
𝑎=𝟎
Speed=max
𝒌
𝑎 = + .𝒙
𝒎
Speed=0
Particle in Simple Harmonic Motion
𝑑𝑥
∵𝑣=
𝑑𝑡
𝑑O𝑥
𝑎= O
𝑑𝑡
𝑑𝑣
𝑎=
𝑑𝑡
𝒌
𝒂𝒙 = − . 𝒙
𝒎
𝑑O𝑥
𝒌
= − .𝒙
O
𝑑𝑡
𝒎
The constant
𝒌
is
𝒎
represented by 𝝎𝟐 to preserve the negative sign
𝝎𝟐
𝒌
=
𝒎
𝒅𝟐 𝒙
𝟐. 𝒙
=
−𝝎
𝒅𝒕𝟐
𝒈𝒓𝒂𝒅𝒊𝒆𝒏𝒕 = −𝝎𝟐
a/ m.s-2
+𝑥š
−𝑥š
x/m
Mathematical Solution of
𝒅𝟐 𝒙
𝒅𝒕𝟐
– Function 𝒙(𝒕) that satisfies this second-order differential equation.
– The trigonometry function sine, and cosine exhibit this behavior.
𝒙 𝒕 = 𝑨 𝐜𝐨𝐬 𝝎𝒕 + ∅
Both ∅, 𝒂𝒏𝒅 𝑨 are constants.
=
𝟐
−𝝎 . 𝒙
Explicitly of This Solution
𝑥 𝑡 = 𝐴 cos 𝜔𝑡 + ∅
𝑑𝑥
∵𝑣=
= −𝐴𝜔 𝑠𝑖𝑛 𝜔𝑡 + ∅
𝑑𝑡
𝑑O𝑥
𝑎 = O = −𝐴𝜔O cos 𝜔𝑡 + ∅
𝑑𝑡
𝒅𝟐 𝒙
𝟐. 𝒙
=
−𝝎
𝒅𝒕𝟐
Graphical Representation
𝑑O𝑥
O cos 𝜔𝑡 + ∅
=
−𝐴𝜔
𝑑𝑡 O
𝒅𝟐 𝒙
𝟐. 𝒙
=
−𝝎
𝒅𝒕𝟐
– A, called the amplitude of the motion, is simply the maximum value
of the position of the particle in either the positive or negative x
direction.
– The constant 𝜔 is called the angular frequency,
– The constant angle ∅ is called the phase constant (initial phase
angle)
– phase constant is determined uniquely by the position and velocity
of the particle at t =0.
– The quantity 𝜔𝑡 + ∅ is called phase of the motion.
Angular frequency 𝝎
– It is a measure of how rapidly the oscillations are occurring.
– The more oscillations per unit time, the higher the value of 𝜔.
– It has units of radians per second.
𝝎𝟐
𝒌
=
𝒎
𝝎=
𝒌
𝒎
𝝎 = 𝟐𝝅𝒇
Q2
𝒇=
𝟏 𝟏
=
𝑻 𝟖
𝟐𝝅
𝝎=
𝟖
𝑥 = 𝐴 𝑠𝑖𝑛 𝜔𝑡
𝟐𝝅 4
3
𝑥 = 1 &times; sin
&times; =
𝟖 3
2
&copy;
𝑫𝒐𝒏 𝒕 𝑭𝒐𝒓𝒈𝒆𝒕 … … … … 𝝅 = 𝟏𝟖𝟎
𝟐𝝅
O
𝑎 = −𝝎 𝑥 = −
𝟖
𝜋O 3
𝑎=−
32
O
3
&times;
2
Equations for velocity
– If we take time t = 0 when the oscillator passes through the middle of the
oscillation with its greatest speed v0, then we can represent the changing
𝑣 = 𝑥š . 𝜔 𝑐𝑜𝑠 𝜔. 𝑡
(1)
𝑡=0
𝒗𝒐 = 𝒙𝒐 . 𝝎
x = x- sin ω. t
x
= sin ω. t
x𝐱
𝐱𝐨
𝟐
= 𝐬𝐢𝐧𝟐 𝛚. 𝐭
(𝟐)
∵ cos O ω. t + sinO ω. t = 1
∴ cos O ω. t = 1 − sinO ω. t
𝐜𝐨𝐬 𝛚. 𝐭 =
𝟏 − 𝐬𝐢𝐧𝟐 𝛚. 𝐭
(𝟑)
From 2, and 3
𝐜𝐨𝐬 𝛚. 𝐭 =
𝐱
𝟏−
𝐱𝐨
𝟐
𝟒
From 1, and 4
x
∵ v = x- . ω 1 −
x-
O
𝒗 = 𝝎 𝒙𝒐 𝟐 − 𝒙𝟐
Kinetic Energy
It is the energy associated to the moving objects.
𝟏
𝑬𝑲 = . 𝒎. 𝒗𝟐
𝟐
𝒗 = 𝝎 𝒙𝒐 𝟐 − 𝒙𝟐
𝟏
𝑬𝑲 = . 𝒎. 𝝎𝟐 . 𝒙𝒐 𝟐 − 𝒙𝟐
𝟐
at time t=0
𝑥 = 𝑥š . 𝑠𝑖𝑛 𝜔. 𝑡
𝑥=0
1
𝐸&para; = . 𝑚. 𝜔O . 𝑥š O
2
Potential Energy:
– it is the work done against the restoring force at a given instant.
1
𝑤𝑜𝑟𝑘 = 𝐸&cedil; = . 𝐹&sup1;LJKš&sup1;n&ordm;&raquo; . 𝑥
2
𝑭𝒓𝒆𝒔𝒕𝒐𝒓𝒊𝒏𝒈 = −𝒎. 𝒂 = 𝒎. 𝝎𝟐 . 𝒙
1
𝐸&cedil; = . 𝑚. 𝜔O . 𝑥. 𝑥
2
𝟏
𝑬𝒑 = . 𝒎. 𝝎𝟐 . 𝒙𝟐
𝟐
Total Energy
– It is the sum of the kinetic energy and potential energy.
– The total energy is conserved according to the law of
conservation of energy in absence of frictional forces.
𝐸KšKml = 𝐸&cedil; + 𝐸&para;
1
1
O
O
O
𝐸KšKml = . 𝑚. 𝜔 . 𝑥š − 𝑥
+ 𝐸&cedil; = . 𝑚. 𝜔O . 𝑥. 𝑥
2
2
𝟏
𝑬𝒕𝒐𝒕𝒂𝒍 = . 𝒎. 𝝎𝟐 . 𝒙𝒐 𝟐
𝟐
Systems in S.H.M
Mass-Spring System
Pendulum
Mass-Spring System
– Since the spring obeys Hook’s law
𝑭 = −𝒌. ∆𝒙
– where k is the spring constant, ∆x is the displacement
– Newton’s second law
𝐅 = 𝐦. 𝐚
𝐦. 𝐚 = −𝐤. ∆𝐱
𝐤
𝐚 = − . ∆𝐱
𝐦
– By comparing this equation with the general formula of the
S.H.M
𝐚 = −𝛚𝟐 . ∆𝐱
𝒌
𝟐
𝝎 =
𝒎
Period and Frequency
ω=
k
m
∵ ω = 2πf
2πf =
k
m
𝟏 𝒌
𝒇=
𝟐𝝅 𝒎
1
f=
T
𝒎
𝑻 = 𝟐𝝅
𝒌
Simple Pendulum
– Tension force in the string T:
𝐓 = −𝐦𝐠 𝒄𝒐𝒔 𝜽
– Restoring force F
F = −mg sin θ
– So we have to keep θ as small as we can at most 10o.
– Because in this range of angles:
𝐬𝐢𝐧 𝛉 𝐝𝐞𝐠𝐫𝐞𝐞 = 𝛉 𝐢𝐧 𝐫𝐚𝐝𝐢𝐚𝐧.
𝐅 = 𝐦. 𝐚
𝐅 = −𝐦𝐠 𝐬𝐢𝐧 𝛉
𝐦. 𝐚 = −𝐦𝐠 𝐬𝐢𝐧 𝛉
𝐚 = −𝐠 𝐬𝐢𝐧 𝛉
From the figure:
∆x
sin θ =
L
𝐱
𝐚 = −𝐠.
𝐋
a = −ωO . ∆x
𝝎𝟐
𝒈
=
𝑳
Period and Frequency
ω=
g
L
ω = 2πf
𝒈
𝟐𝛑𝐟 =
𝑳
𝟏 𝒈
𝒇=
𝟐𝝅 𝑳
1
T=
f
𝑳
𝑻 = 𝟐𝝅
𝒈
Damped Oscillations
– It is the gradual decrease in the energy of the oscillating body because of
frictional force.
𝟏
𝑬𝒕𝒐𝒕𝒂𝒍 = . 𝒎. 𝝎𝟐 . 𝒙𝒐 𝟐
𝟐
Car Damper
Oil or
other viscous
fluid
Coil spring
Piston
with holes
a
b
Shock absorber
Q1
Propagation of a Disturbance
Transverse
• The disturbance is in a direction perpendicular to
the propagation direction of the wave.
Longitudinal
• The disturbance is in a direction parallel to the
propagation direction of the wave.
Combination of Transverse and
Longitudinal Displacements
– Surface-water waves are a good example.
– When a water wave travels on the surface of deep
water, elements of water at the surface move in
nearly circular paths.
– The disturbance has both transverse and longitudinal
components.
– The transverse displacements seen represent the
variations in vertical position of the water elements.
– The longitudinal displacements represent elements of
water moving back and forth in a horizontal direction.
The Three-Dimensional Waves
– The slower transverse waves, called S waves, with “S” standing for secondary, travel through
the Earth at 4 to 5 km/s near the surface.
– By recording the time interval between the arrivals of these two types of waves at a
seismograph, the distance from the seismograph to the point of origin of the waves can be
determined.
– This distance is the radius of an imaginary sphere centered on the seismograph.
– The origin of the waves is located somewhere on that sphere.
– The imaginary spheres from three or more monitoring stations located far apart from one
another intersect at one region of the Earth, and this region is where the earthquake
occurred.
– That travel out from a point under the Earth’s surface at which an earthquake occurs.
– Both types, transverse and longitudinal are formed.
– The longitudinal waves are the faster than the transverse waves, and traveling at
speeds in the range of 7 to 8 km/s near the surface.
– These longitudinal waves are called P waves, with “P” standing for primary, because
they travel faster than the transverse waves and arrive first at a Seismo-graph (a device
used to detect waves due to earthquakes).
Wave Function
– Figure represents the shape and position of the pulse at time t = 0.
– Mathematical function:
𝑦 𝑥, 0 = 𝑓(𝑥).
– This function describes the transverse position y of the element of
the string located at each value of x at time t = 0.
– The speed of the pulse is v.
– The pulse has traveled to the right a 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑣𝑡 at the time t.
Wave Function
– We assume the shape of the pulse does not change with time.
– At time t, the shape of the pulse is the same as it was at time t = 0
– Consequently, an element of the string at x at this time has the same y position as an
element located at x - vt had at time t = 0
𝑦 𝑥, 𝑡 = 𝑦(𝑥 − 𝑣𝑡).
In general,
𝒚 𝒙, 𝒕 = 𝒇(𝒙 − 𝒗𝒕).
Wave function, depends on the two variables x and t
Waveform
– If t is fixed (as, for example, in the case of taking a snapshot of the pulse),
– the wave function y(x), sometimes called the waveform, defines a curve representing
the geometric shape of the pulse at that time.
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