chapter 13. simple harmonic motion a single sequence of moves that constitutes the repeated unit in...
TRANSCRIPT
Harmonic MotionChapter 13
Simple Harmonic Motion
A single sequence of moves that constitutes the repeated unit in a periodic motion is called a cycle
The time it takes for a system to complete a cycle is a period (T)
Simple Harmonic MotionThe period is the number of units of time per
cycle; the reciprocal of that—the number of cycles per unit of time—is known as the frequency (f).
The SI unit of frequency is the hertz (Hz), where 1 Hz = 1 cycle/s = 1 s-1
Amplitude (A) is the maximum displacement of an
object in SHM
Tf
1
Simple Harmonic MotionOne complete orbit (one
cycle) object sweeps through 2p rad
f - number of cycles per second
Number of radians it moves through per second is 2pf - that's angular speed ( )w T
f 2
2
w - angular frequency Sinusoidal motion (harmonic) with a single
frequency - known as simple harmonic motion (SHM)
Displacement in SHM
txAx coscos max
Velocity in SHM
tAvx sin
2max )/(1sin AxvtAvx
vmax = Aw
Acceleration in SHM
tAax cos2xax
2
The acceleration of a simple harmonic oscillator is proportional to its displacement
2max Aa
Example 1A spot of light on the screen of a computer
is oscillating to and fro along a horizontal straight line in SHM with a frequency of 1.5 Hz. The total length of the line traversed is 20 cm, and the spot begins the process at the far right. Determine
(a) its angular frequency, (b) its period, (c) the magnitude of its maximum velocity,
and (d) the magnitude of its maximum
acceleration, (e) Write an expression for x and find the
location of the spot at t = 0.40 s.
Problem
A point at the end of a spoon whose handle is clenched between someone’s teeth vibrates in SHM at 50Hz with an amplitude of 0.50cm. Determine its acceleration at the extremes of each swing.
Equilibrium
The state in which an elastic or oscillating system most wants to be in if undisturbed by outside forces.
Elastic Restoring ForceWhen a system oscillates naturally it moves against a restoring force that returns it to its undisturbed equilibrium condition
A "lossless" single-frequency ideal vibrator is known as a simple harmonic oscillator.
An Oscillating SpringIf a spring with a mass attached
to it is slightly distorted, it will oscillate in a way very closely resembling SHM.
Force exerted by an elastically stretched spring is the elastic restoring force F, = -ks.
Resulting acceleration ax = -(k/m)x
F is linear in x; a is linear in x - hallmark of SHM
Frequency and Period
Simple harmonic oscillator
Shown every ¼ cycles for 2 cycles
Relationship between x, vx, t, and T
Hooke’s Law
Beyond being elastic, many materials deform in proportion to the load they support - Hooke's Law
sF
Hooke’s Law
The spring constant or elastic constant k - a measure of the stiffness of the object being deformed
k has units of N/m
ksF
Hooke’s Law
k has units of N/m
Frequency and Period
w0 - the natural angular frequency, the specific frequency at which a physical system oscillates all by itself once set in motion
natural angular frequency
and since w0 = 2pf0
natural linear frequency
Since T= 1/f0
Period
m
k0
m
kf
21
0
k
mT 2
Resonance vs. Damping• If the frequency of the disturbing force equals the natural frequency of the system, the amplitude of the oscillation will increase—RESONANCE
• If the frequency of the periodic force does NOT equal the natural frequency of the system, the amplitude of the oscillation will decrease--DAMPING
Example 2A 2.0-kg bag of candy is hung on a vertical,
helical, steel spring that elongates 50.0 cm under the load, suspending the bag 1.00 m above the head of an expectant youngster. The candy is pulled down an additional 25.0 cm and released. How long will it take for the bag to return to a height of 1.00 m above the child?
The PendulumThe period of a
pendulum is independent of the mass and is determined by the square root of its length
L
gf
21
0
g
LT 2
Example 3How long should a pendulum be if it is to
have a period of 1.00 s at a place on Earth where the acceleration due to gravity is 9.81 m/s2?
Problem 2What would the length of a pendulum need to
be on Jupiter in order to keep the same time as a clock on Earth? gJupiter = 25.95m/s2
Tf
1
Tf
22
txAx coscos max
vmax = Aw 2max Aa
ksF m
k0
m
kf
21
0
k
mT 2
When “f” is known
When “f” is NOT known
L
gf
21
0
g
LT 2