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Oscillations in the spring-mass system
Maximum speed, maximum kinetic energy
Maximum force, maximum acceleration
Will the period change if the amplitude changes?
period
• Frequency and period are not dependend on the amplitude!
• What are the implications for musical instruments?
• Pitch is not dependent on how much the string is plucked (for example)
Does the period depend on the mass?
Which force is acting on both masses?
F kx
Which acceleration does each mass experience?
ma kxkx
am
The smaller mass accelerates at a higher rate: it is faster to move through the cycle.
It will oscillate at a smaller period (faster).
Properties of the spring and period
stiff soft
Large spring constant k1
Small spring constant k2
kxa
m
Large k means high acceleration
High acceleration means a cycle is completed fast
High k <-> low period
Factors influencing the period:
Influence Period Frequency
Mass m
Stiffness k
Amplitude A No influence No influence
Frequency and Period
1
2
kf
m
2m
k
Simple harmonic motion
What is influenced by the amplitude?
21
2E KE EPE kA
What happens if energy gets lost due to friction?
• A) the frequency decreases
• B) the period decreases
• C) the amplitude decreases
• D) all of the above
The amplitude decreases.
Frequency and period stay the same.
Damped harmonic motion
Example
Clink of a coffee cup
Example
• Finger flicking Short, hard to assign pitch
Driven oscillator
1spring and mass:
2
kf
m
Slow driving frequency:f0 < f
Mass moves in rhythmwith the driver.
Driven oscillator
1spring and mass:
2
kf
m
Fast driving frequency: f0 > f
The mass is practicallynot moving.
Driven oscillator
1spring and mass:
2
kf
m
Resonance:
f0 = f
Amplitude grows very much!
Amplitude versus driving frequency
Resulting amplitude
Driving frequencyNatural frequencyof oscillator
The shocks of an automobile are springs. Which of the following would
be desirable?• A) resonance with the bumps on the road,
and little damping
• B) high damping and high spring constant
• C) high damping and very low spring constant
• D) resonance with the bumps on the road, and high damping
A) resonance with the bumps on the road, and little damping
B) high damping and high spring constant
One bump, high spring constant, medium damping
One bump, high damping, medium spring constant
Several bumps, high damping, medium spring constant
Answer C: similar effects
D) resonance with the bumps on the road, and high damping
Multiple bumps
Examples of other oscillators
• Pendulum (small amplitude)
1
2
gf
L
L
Examples of other oscillators• Air-filled piston
1 1.4
2
Af
mL
m
L
A
AtmosphericPressure P0
P0
m
AtmosphericPressure P0
P0+p
m
AtmosphericPressure P0
P0-p
F=pA
Examples of other oscillators• Helmholtz Resonator
1
2
Af
VLL
V
A