Introductory Video: Simple Harmonic Motion Simple Harmonic Motion.
Simple Harmonic Motion Wenny Maulina Simple harmonic motion Simple harmonic motion (SHM) Solution:...
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Transcript of Simple Harmonic Motion Wenny Maulina Simple harmonic motion Simple harmonic motion (SHM) Solution:...
Simple Harmonic Motion
Wenny Maulina
Simple harmonic motion
Simple harmonic motion (SHM)
• Solution:
)cos()( tAtx
• What is SHM?
A simple harmonic motion is the motionof an oscillating system which satisfiesthe following condition:
1. Motion is about an equilibrium position at which point no net force acts on the
system.2. The restoring force is proportional to and oppositely directed to the displacement.3. Motion is periodic.
t=0t=- /f w
Acosf
f= /(2 )w p
By Dr. Dan Russell, Kettering University
=w w0; =w 2w0 ; =w 3w0
Simple Harmonic Motion, SHMSimple harmonic motion is periodic motion in the absence of friction and produced by a restoring force that is directly proportional to the displacement and oppositely directed.
A restoring force, F, acts in the direction opposite the displacement of the oscillating body.
F = -kx
A restoring force, F, acts in the direction opposite the displacement of the oscillating body.
F = -kx
x F
Oscillations of a SpringHooke’s Law states Fs = -kx
Fs is the restoring force.It is always directed toward the equilibrium position.Therefore, it is always opposite the displacement from equilibrium.
k is the force (spring) constant.x is the displacement.
Oscillations of a Spring In a, the block is displaced to the right of x = 0.
The position is positive. The restoring force is directed to the left (negative).
In b, the block is at the equilibrium position.
x = 0 The spring is neither stretched nor
compressed. The force is 0.
In c, the block is displaced to the left of x = 0.
The position is negative. The restoring force is directed to the right (positive).
6
Example
A 0.42-kg block is attached to the end of a horizontal ideal
spring and rests on a frictionless surface. The block is pulled so
that the spring stretches by 2.1 cm relative to its length. When
the block is released, it moves with an acceleration of 9.0 m/s2.
What is the spring constant of the spring?
7
2.1cm
kx = ma
2/0.942.0100
1.2smk
mNk /1801001.2
0.942.0
Displacement in SHM
• Displacement is positive when the position is to the right of the equilibrium position (x = 0) and negative when located to the left.
• The maximum displacement is called the amplitude A.
m
x = 0 x = +Ax = -A
x
Velocity in SHM
m
x = 0 x = +Ax = -A
v (+)
• Velocity is positive when moving to the right and negative when moving to the left.
• It is zero at the end points and a maximum at the midpoint.
v (-)
Acceleration in SHM
m
x = 0 x = +Ax = -A
• Acceleration is in the direction of the restoring force. (a is positive when x is negative, and negative when x is positive.)
• Acceleration is a maximum at the end points and it is zero at the center of oscillation.
+x-a
-x+a
F ma kx
Acceleration vs. Displacement
m
x = 0 x = +Ax = -A
x va
Given the spring constant, the displacement, and the mass, the acceleration can be found from:
or
Note: Acceleration is always opposite to displacement.
F ma kx kx
am
Simple harmonic motion
Displacement, velocity and acceleration in SHM
• Displacement
)cos()( tAtx
• Velocity
)sin()(
)( tAdt
tdxtv
)cos()(
)( 2 tAdt
tdvta
• Acceleration
0
The diaphragm of a loudspeaker moves back and forth in simple
harmonic motion to create sound. The frequency of the motion is f
= 1.0 kHz and the amplitude is A = 0.20 mm.
(a)What is the maximum
speed of the diaphragm?
(b)Where in the motion does
this maximum speed
occur?
Example
(b) The speed of the diaphragm is zero when the diaphragm
momentarily comes to rest at either end of its motion: x = +A
and x = –A. Its maximum speed occurs midway between these
two positions, or at x = 0 m.
(a)
A loudspeaker diaphragm is
vibrating at a frequency of f =
1.0 kHz, and the amplitude of
the motion is A = 0.20 mm.
(a)What is the maximum
acceleration of the
diaphragm, and
(b)where does this maximum
acceleration occur?
Example
(b) the maximum acceleration occurs at x = +A and x = –A
(a)
The drawing shows plots
of the displacement x
versus the time t for three
objects undergoing simple
harmonic motion. Which
object, I, II, or III, has the
greatest maximum
velocity?
Example
The cone of a loudspeaker oscillates in
SHM at a frequency of 262 Hz. The
amplitude at the center of the cone is A =
1.5 x 10-4 m, and at t = 0, x = A. (a) What
equation describes the motion of the center
of the cone? (b) What are the velocity and
acceleration as a function of time? (c)
What is the position of the cone at t = 1.00
ms (= 1.00 x 10-3 s)?
Example
Solution:
.1650cosm105.1
)2cos()(
,2cos)0(
),cos()(
rad/s,1650Hz26222 a.
4 t
tAtx
AAx
tAtx
f
Solution:
m.102.1
s1000.1rad/s1650cosm105.1
ms,00.1at c.
1650cosm/s410)(
;1650sinm/s25.0)( b.
5
34
2
x
t
tdt
dvta
tdt
dxtv
A spring stretches 0.150 m when a 0.300-kg mass is gently attached to it. The spring is then set up horizontally with the 0.300-kg mass resting on a frictionless table. The mass is pushed so that the spring is compressed 0.100 m from the equilibrium point, and released from rest. Determine: (a) the spring stiffness constant k and angular frequency ω; (b) the amplitude of the horizontal oscillation A; (c) the magnitude of the maximum velocity vmax; (d) the magnitude of the maximum acceleration amax of the mass; (e) the period T and frequency f; (f) the displacement x as a function of time; and (g) the velocity at t = 0.150 s.
Example
Solution:
frequencyangular :ω
nt)displaceme (maximum amplitude:Am
kω whereωt,sin A x :solution simple a
kxdt
xdm
:Motion ofEquation
2
2
Oscillations of a Spring
F = - k x
k
x
m
F = - k x
x
Displacement : x = L θ
Returning force : F = - mg sin θ
Newton Law : F = m d2θ/dt2
Small oscillation approximation : sin θ θ
Acceleration : d2θ/dt2 = - (g/L) x
Solutions : Y = A sin wt , where w2 = g/L
Y = A cos wt
Y = A eiwt
Simple Pendulum
Taylor series :sin𝜃=𝜃−𝜃3
3 !+ 𝜃
5
5 !−⋯
• The fact that the velocity is zero at maximum displacement in simple harmonic motion and is a maximum at zero displacement illustrates the important concept of an exchange between kinetic and potential energy.
• If no energy is dissipated then all the potential energy becomes kinetic energy and vice versa, so that the values of (a) the total energy at any time, (b) the maximum potential energy and (c) the maximum kinetic energy will all be equal; that is
Energy in SHM
UKE
Energy in SHM
Energy conservation
UKE Energy conservation in a SHM
dx
dUkxFs
No friction22
2
1 ;
2
1kxUmvK
const. 2
1
2
1 22 kxmvE
EkxmvkA 222
2
1
2
1
2
1
22 xAm
kv
BTW:
2
0 2
1 kxdxFUU
x
s
w2
Energy in SHM
Energy conservation in a SHM
const. 2
1
2
1 22 kxmvE
ene
rgy
ene
rgy
distance from equilibrium pointTime
E
kinetic energy
potential energy
This graph shows the potential energy function of a spring. The total energy is constant.
Energy in SHM
An object of mass m = 0.200 kg that is vibrating on a horizontal frictionless table. The spring has a spring constant k = 545 N/m. It is stretched initially to x0 = 4.50 cm and then released from rest (see part A of the drawing). Determine the final translational speed vf of the object when the final displacement of the spring is (a) xf = 2.25 cm and (b) xf = 0 cm.
Example
30
20
22
2
1
2
1
2
1kxkxmv ff
)( 220 ff xx
m
kv
0EE f
20
20
22
2
1
2
1
2
1
2
1kxmvkxmv ff
31
(a) Since x0 = 0.0450 m and xf = 0.0225 m,
(b) When x0 = 0.0450 m and xf = 0 m,
What must be the length of a simple pendulum for a clock which has a period of two seconds (tick-tock)?
L
Example
What must be the length of a simple pendulum for a clock which has a period of two seconds (tick-tock)?
2L
Tg
L
L = 0.993 m
Example
L
L
Many tall building have mass dampers, which are anti-sway devices to prevent them from oscillating in a wind. The device might be a block oscillating at the end of a spring and on a lubricated track. If the building sways, say eastward, the block also moves eastward but delayed enough so that when it finally moves, the building is then moving back westward. Thus, the motion of the oscillator is out of step with the motion of the building. Suppose that the block has mass m = 2.72 x 105 kg and is designed to oscillate at frequency f = 10.0 Hz and with amplitude xm = 20.0 cm.
(a) What is the total mechanical energy E of the spring-block system?
Example
Damped harmonic motion is harmonic motion with a frictional or drag force.
DAMPED OSCILLATION
There are systems in which damping is unwanted, such as clocks and watches.
Then there are systems in which it is wanted, and often needs to be as close to critical damping as possible, such as automobile shock absorbers and earthquake protection for buildings.
The friction reduces the mechanical energy of the system as time passes, and the motion is said to be damped.
The smallest degree of damping that completely eliminates the oscillations is termed “critical damping,” and the motion is said to be critically damped.
When the damping exceeds the critical value, the motion is said to be overdamped. In contrast, when the damping is less than the critical level, the motion is said to be underdamped.
DAMPED OSCILLATION
DAMPED OSCILLATION
When a periodically varying driving force with angular frequency is applied to a damped harmonic oscillator, the resulting motion is called a forced oscillation.
There are two frequencies involved in a forced oscillator:I. ω0, the natural angular frequency of the oscillator, without
the presence of any external force, andII. ω, the angular frequency of the applied external force.
If the frequency is the same as the natural frequency, the amplitude can become quite large. This is called resonance.
Forced Oscillation
The sharpness of the resonant peak depends on the damping. If the damping is small (A) it can be quite sharp; if the damping is larger (B) it is less sharp.
Like damping, resonance can be wanted or unwanted. Musical instruments and TV/radio receivers depend on it.
Forced Oscillation
Please make a paper about Standing waves and Travelling waves
ASSIGNMENT 1Should Be Submitted Next Week